SAT Math Fundamentalsthe reference sheet

1.1 Addition and Subtraction of Rational Numbers

1. 1.1.1

   Subtracting a Positive Number From a Smaller Positive Number

   The answer is negative. Find the gap, then make it negative.

   e.g. $3 - 8 = -5$

   $\displaystyle a - b = -(b - a) \quad \text{when } b > a$
2. 1.1.2

   Subtracting a Positive Number From a Negative Number

   Move further left. Add the sizes, keep the negative sign.

   e.g. $-4 - 6 = -10$

   $\displaystyle -a - b = -(a + b)$
3. 1.1.3

   Subtracting a Positive Fraction From a Smaller Fraction

   Get a common denominator, then subtract the same way.

   $\displaystyle \frac{1}{4} - \frac{2}{3} = \frac{3}{12} - \frac{8}{12} = -\frac{5}{12}$
4. 1.1.4

   Subtracting a Positive Decimal From a Smaller Decimal

   Line up the decimal points, subtract larger minus smaller, attach the negative.

   $\displaystyle 1.2 - 3.75 = -2.55$
5. 1.1.5

   Adding a Positive Number to a Negative Number

   Signs differ: subtract the sizes and keep the sign of the bigger size.

   $\displaystyle -7 + 3 = -4$

   $\displaystyle -2 + 9 = 7$
6. 1.1.6

   Adding a Negative Number

   Adding a negative is the same as subtracting.

   e.g. $5 + (-8) = -3$

   $\displaystyle a + (-b) = a - b$
7. 1.1.7

   Subtracting a Negative Number

   Subtracting a negative is the same as adding.

   e.g. $-2 - (-6) = 4$

   $\displaystyle a - (-b) = a + b$
8. 1.1.8

   Additive Inverses of Numbers

   A number and its opposite add to zero. They sit the same distance from 0 on opposite sides.

   $\displaystyle a + (-a) = 0$
9. 1.1.9

   Interpreting Additive Opposites and Addition Using Absolute Value

   Absolute value is distance from zero, so it is never negative. $a + b$ lands $\lvert b \rvert$ units from $a$.

   $\displaystyle \lvert a \rvert = \text{distance from } 0$

   $\displaystyle \lvert -5 \rvert = \lvert 5 \rvert = 5$
10. 1.1.10

    Solving Problems Using Addition and Subtraction of Rational Numbers

    Model it on a number line: right or up is +, left or down is −. Temperatures, elevation, bank balances.

    e.g. 3°F falls 10° → $3 - 10 = -7$°F

1.2 Multiplication and Division of Rational Numbers

1. 1.2.1

   Multiplying Positive Numbers With Negative Numbers

   Different signs give a negative.

   e.g. $4 \times (-3) = -12$

   $\displaystyle (+)(-) = (-)$
2. 1.2.2

   Multiplying Negative Numbers

   Same signs give a positive. An even count of negatives is positive; an odd count is negative.

   e.g. $(-2)(-3)(-1) = -6$

   $\displaystyle (-)(-) = (+)$
3. 1.2.3

   Dividing With Negative Numbers

   Same sign rules as multiplication.

   $\displaystyle \frac{-12}{4} = -3$

   $\displaystyle \frac{-12}{-4} = 3$
4. 1.2.4

   Equivalent Fractions With Negative Numbers

   One negative can sit in any of three places. Two negatives cancel.

   $\displaystyle -\frac{a}{b} = \frac{-a}{b} = \frac{a}{-b}$

   $\displaystyle \frac{-a}{-b} = \frac{a}{b}$
5. 1.2.5

   Dividing With Negative Decimals and Fractions

   Settle the sign first, then divide the sizes. For fractions, multiply by the reciprocal.

   e.g. $-\tfrac34 \div \tfrac12 = -\tfrac34 \cdot 2 = -\tfrac32$

6. 1.2.6

   Reciprocals of Rational Numbers

   Flip the fraction. A number times its reciprocal is 1. The sign stays; zero has no reciprocal.

   e.g. reciprocal of $-\tfrac25$ is $-\tfrac52$

   $\displaystyle \frac{a}{b} \cdot \frac{b}{a} = 1$
7. 1.2.7

   Multiplying and Dividing With Zero

   Anything times zero is zero. Zero divided by a number is zero. Dividing by zero is undefined.

   $\displaystyle a \times 0 = 0$

   $\displaystyle \frac{0}{a} = 0$

   $\displaystyle \frac{a}{0} \text{ is undefined}$
8. 1.2.8

   Fractions of Fractions

   “Of” means multiply. Multiply across.

   e.g. $\tfrac23$ of $\tfrac34$ $= \tfrac{6}{12} = \tfrac12$

   $\displaystyle \frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$
9. 1.2.9

   Solving Problems Using Multiplication and Division of Rational Numbers

   Decide the operation, settle the sign, then compute.

   e.g. Drops 1.5 m per hour for 4 hours → $-1.5 \times 4 = -6$ m

10. 1.2.10

    Solving Problems Using the Four Operations on Rational Numbers

    Use order of operations and the sign rules together.

    e.g. $-3 + 4 \times (-2) = -3 - 8 = -11$

1.3 Exponents

1. 1.3.1

   Squaring Rational Numbers

   Multiply the number by itself. A negative squared is positive.

   $\displaystyle x^2 = x \cdot x$

   $\displaystyle \left(-\tfrac{2}{3}\right)^2 = \tfrac{4}{9}$
2. 1.3.2

   Cubing Rational Numbers

   Three copies multiplied. A cube keeps the sign.

   $\displaystyle x^3 = x \cdot x \cdot x$

   $\displaystyle (-2)^3 = -8$
3. 1.3.3

   Exponents With Rational Bases

   Raise the top and the bottom. Watch the parentheses: the exponent only touches what it is attached to.

   $\displaystyle \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$

   $\displaystyle -3^2 = -9 \quad (-3)^2 = 9$
4. 1.3.4

   The Zeroth Power

   Any nonzero number to the zero power is 1.

   e.g. $(-7)^0 = 1$

   $\displaystyle a^0 = 1 \quad (a \ne 0)$
5. 1.3.5

   Powers of Negative One

   Even power gives 1, odd power gives −1.

   $\displaystyle (-1)^n = \begin{cases} 1 & n \text{ even} \\ -1 & n \text{ odd} \end{cases}$
6. 1.3.6

   Negative Exponents

   A negative exponent means reciprocal. It does not make the number negative.

   e.g. $2^{-3} = \tfrac18$

   $\displaystyle a^{-n} = \frac{1}{a^n}$

   $\displaystyle \left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n}$
7. 1.3.7

   The Product Rule for Exponents

   Same base, multiplying: add the exponents.

   e.g. $x^3 \cdot x^4 = x^7$

   $\displaystyle a^m \cdot a^n = a^{m+n}$
8. 1.3.8

   The Quotient Rule for Exponents

   Same base, dividing: subtract the exponents.

   e.g. $\tfrac{x^7}{x^2} = x^5$

   $\displaystyle \frac{a^m}{a^n} = a^{m-n}$
9. 1.3.9

   The Power Rule for Exponents

   Power of a power: multiply the exponents.

   e.g. $(x^2)^5 = x^{10}$

   $\displaystyle (a^m)^n = a^{mn}$
10. 1.3.10

    The Power of Product Rule for Exponents

    The exponent goes to every factor inside.

    e.g. $(2x)^3 = 8x^3$

    $\displaystyle (ab)^n = a^n b^n$
11. 1.3.11

    The Power of Quotient Rule for Exponents

    The exponent goes to the top and the bottom.

    $\displaystyle \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$
12. 1.3.12

    Combining the Rules of Exponents

    Work inside parentheses, apply outside powers, combine like bases, finish with positive exponents.

    e.g. $\dfrac{(x^2)^3 \cdot x}{x^4} = \dfrac{x^7}{x^4} = x^3$

1.4 Radicals

1. 1.4.1

   The Square Root of a Perfect Square

   $\sqrt{\ }$ asks “what nonnegative number squared gives this?” Know squares through 15² = 225.

   e.g. $\sqrt{49} = 7$, $\sqrt{\tfrac{9}{16}} = \tfrac34$

   $\displaystyle \sqrt{a^2} = a \quad (a \ge 0)$
2. 1.4.2

   Squaring a Square Root

   Squaring undoes the square root.

   e.g. $(\sqrt{7})^2 = 7$

   $\displaystyle \left(\sqrt{a}\right)^2 = a \quad (a \ge 0)$
3. 1.4.3

   The Cube Root of a Perfect Cube

   Asks “what cubed gives this?” Negatives are allowed and stay negative.

   $\displaystyle \sqrt[3]{a^3} = a$

   $\displaystyle \sqrt[3]{-27} = -3$
4. 1.4.4

   Surds

   A root that never comes out to a whole number or fraction, like $\sqrt2$. It is irrational, so leave it in exact form.

   e.g. $\sqrt2, \sqrt3, \sqrt5$ are surds; $\sqrt9$ is not

5. 1.4.5

   Simplifying Expressions With Surds

   Pull out the largest perfect-square factor.

   $\displaystyle \sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}$
6. 1.4.6

   Evaluating Numerical Expressions Containing Radicals

   A radical acts like parentheses: simplify everything under it first.

   e.g. $3 + \sqrt{25 - 9} = 3 + 4 = 7$

7. 1.4.7

   Adding and Subtracting Radicals

   Only like radicals (same number under the root) combine. Simplify each one first.

   $\displaystyle 3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2}$

   $\displaystyle \sqrt{8} + \sqrt{2} = 2\sqrt2 + \sqrt2 = 3\sqrt{2}$
8. 1.4.8

   Rationalizing Denominators

   Clear a root from the bottom by multiplying top and bottom by that root.

   e.g. $\tfrac{6}{\sqrt3} = \tfrac{6\sqrt3}{3} = 2\sqrt3$

   $\displaystyle \frac{a}{\sqrt{b}} = \frac{a\sqrt{b}}{b}$
9. 1.4.9

   Rationalizing Denominators With Two Terms

   Multiply by the conjugate: the same two terms with the middle sign flipped. The difference of squares clears the root.

   e.g. $\tfrac{1}{3 + \sqrt2} = \tfrac{3 - \sqrt2}{7}$

   $\displaystyle \frac{1}{a + \sqrt{b}} \cdot \frac{a - \sqrt{b}}{a - \sqrt{b}} = \frac{a - \sqrt{b}}{a^2 - b}$
10. 1.4.10

    Radicals as Fractional Exponents

    A root is a fractional power. The bottom of the fraction is the root.

    e.g. $8^{2/3} = (\sqrt[3]{8})^2 = 4$

    $\displaystyle \sqrt[n]{a} = a^{1/n}$

    $\displaystyle \sqrt[n]{a^m} = a^{m/n}$
11. 1.4.11

    The Product Rule for Radicals

    The root of a product is the product of the roots.

    e.g. $\sqrt{3}\sqrt{12} = \sqrt{36} = 6$

    $\displaystyle \sqrt{ab} = \sqrt{a}\,\sqrt{b} \quad (a, b \ge 0)$
12. 1.4.12

    The Quotient Rule for Radicals

    The root of a quotient is the quotient of the roots.

    e.g. $\sqrt{\tfrac{25}{4}} = \tfrac52$

    $\displaystyle \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$
13. 1.4.13

    Applying the Distributive Law to Radical Expressions

    Distribute or FOIL as usual, multiply radicals together, then simplify. Conjugates eliminate the radical.

    e.g. $\sqrt3(\sqrt6 + 2) = \sqrt{18} + 2\sqrt3 = 3\sqrt2 + 2\sqrt3$

    $\displaystyle \sqrt a\,(\sqrt b + \sqrt c) = \sqrt{ab} + \sqrt{ac}$

    $\displaystyle (\sqrt a + b)(\sqrt a - b) = a - b^2$
14. 1.4.14

    Comparing Radicals

    Move each coefficient inside the root by squaring it, then compare the numbers under the roots. A larger radicand means a larger root.

    e.g. $3\sqrt5 = \sqrt{45}$ and $5\sqrt2 = \sqrt{50}$, so $5\sqrt2$ is larger

    $\displaystyle a\sqrt b = \sqrt{a^2 b} \quad (a \ge 0)$

1.5 Order of Operations

1. 1.5.1

   Natural Numbers, Integers, and Rational Numbers

   Each set sits inside the next. Rational means it can be written as a fraction of integers.

   $\displaystyle \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q}$

   $\displaystyle \text{rational} = \frac{a}{b},\ a, b \text{ integers},\ b \ne 0$
2. 1.5.2

   Rational Numbers as Finite or Repeating Decimals

   Every rational number either ends or repeats. It ends when the simplified denominator has only 2s and 5s as prime factors.

   $\displaystyle \frac{3}{8} = 0.375$

   $\displaystyle \frac{1}{3} = 0.\overline{3}$

   $\displaystyle \frac{2}{11} = 0.\overline{18}$
3. 1.5.3

   Evaluating Rational Number Expressions Without Exponents

   Parentheses first. Then multiply and divide left to right. Then add and subtract left to right.

   e.g. $8 - 6 \div 3 \times 2 = 8 - 4 = 4$

4. 1.5.4

   Evaluating Expressions With Positive Integer Exponents

   PEMDAS: exponents come right after parentheses.

   e.g. $2 + 3 \cdot 4^2 = 2 + 48 = 50$

   $\displaystyle \text{P} \to \text{E} \to \text{MD} \to \text{AS}$
5. 1.5.5

   Evaluating Expressions With Integer Exponents

   Handle negative exponents at the exponent step by turning them into reciprocals.

   e.g. $2 + 3 \cdot 2^{-1} = 2 + \tfrac32 = \tfrac72$

1.6 Algebraic Expressions

1. 1.6.1

   Introduction to Algebraic Expressions

   Numbers, variables, and operations with no equals sign. A variable stands for a number you don't know yet.

   $\displaystyle 3x + 5$
2. 1.6.2

   Constructing Algebraic Expressions

   Translate the words. Sum +, difference −, product ×, quotient ÷. “Less than” and “more than” flip the order.

   e.g. “5 less than twice $n$” → $2n - 5$

3. 1.6.3

   Evaluating Linear Expressions

   Substitute the value in parentheses, then follow order of operations.

   $\displaystyle 3x - 4,\ x = -2$

   $\displaystyle 3(-2) - 4 = -10$
4. 1.6.4

   Evaluating Rational Expressions

   Work out the top and the bottom separately, then divide. The bottom can never be 0.

   e.g. $\dfrac{x+4}{x-1}$ at $x=3$ → $\tfrac72$

5. 1.6.5

   Identifying Terms in Linear Expressions

   Terms are the pieces separated by + or −. The sign in front belongs to the term.

   e.g. $4x - 7y + 2$ has terms $4x$, $-7y$, $2$

6. 1.6.6

   Identifying Coefficients and Constants

   The coefficient is the number multiplying a variable. The constant has no variable. A lone $x$ has coefficient 1.

   e.g. In $-5x + 8$: coefficient $-5$, constant $8$

7. 1.6.7

   The Greatest Common Factor of Two Linear Expressions

   Factor each fully, then keep what they share.

   $\displaystyle 6x + 9 = 3(2x + 3)$

   $\displaystyle 12x + 18 = 6(2x + 3)$

   $\displaystyle \text{GCF} = 3(2x + 3)$

1.7 Simplifying Linear Expressions

1. 1.7.1

   Identifying Like Terms in Linear Expressions

   Like terms have the same variable to the same power. Constants are like each other.

   e.g. $3x$ and $-5x$: like. $3x$ and $3y$: not.

2. 1.7.2

   Collecting Like Terms in Linear Expressions

   Add the coefficients of like terms; the variable part stays.

   $\displaystyle 3x + 5 - x + 2 = 2x + 7$
3. 1.7.3

   Applying the Distributive Law With Linear Expressions

   Multiply the outside number by every term inside.

   e.g. $4(x + 3) = 4x + 12$

   $\displaystyle a(b + c) = ab + ac$
4. 1.7.4

   The Distributive Law With Three Terms

   Same rule, one more term.

   e.g. $2(x - y + 5) = 2x - 2y + 10$

   $\displaystyle a(b + c + d) = ab + ac + ad$
5. 1.7.5

   Distributing the Negative Sign in a Linear Expression

   A minus in front flips the sign of every term inside.

   $\displaystyle -(a - b + c) = -a + b - c$
6. 1.7.6

   Simplifying Linear Expressions Using the Distributive Law

   Distribute first, then collect like terms.

   $\displaystyle 2(x + 3) + 4x = 6x + 6$
7. 1.7.7

   Simplifying Linear Expressions When a Group of Terms Is Subtracted

   Distribute the −1 (or −k) to every term in the group.

   $\displaystyle 5x - (2x - 3) = 5x - 2x + 3 = 3x + 3$
8. 1.7.8

   Simplifying Linear Expressions Containing Fractions

   Distribute, then combine with a common denominator.

   $\displaystyle \tfrac{1}{2}(4x - 6) = 2x - 3$

   $\displaystyle \tfrac{x}{2} + \tfrac{x}{3} = \tfrac{5x}{6}$
9. 1.7.9

   Factoring Numerical Expressions

   Pull out the GCF. Factoring is distributing in reverse.

   $\displaystyle 12 + 18 = 6(2 + 3)$
10. 1.7.10

    Factoring Linear Expressions

    Find the GCF of the terms and write it outside.

    $\displaystyle ab + ac = a(b + c)$

    $\displaystyle 8x - 12 = 4(2x - 3)$

2.8 Solving Linear Equations

1. 2.8.1

   Solving Linear Equations by Trial and Error

   Guess a value, substitute, check whether both sides match, adjust the guess.

   e.g. $3x + 1 = 13$: try 4 → $13$ ✓

2. 2.8.2

   Solving One-Step Addition and Subtraction Equations

   Do the opposite operation to both sides.

   $\displaystyle x + a = b \Rightarrow x = b - a$

   $\displaystyle x - a = b \Rightarrow x = b + a$
3. 2.8.3

   Solving One-Step Multiplication and Division Equations

   Divide to undo multiplying, multiply to undo dividing.

   $\displaystyle ax = b \Rightarrow x = \frac{b}{a}$

   $\displaystyle \frac{x}{a} = b \Rightarrow x = ab$
4. 2.8.4

   Solving Two-Step Equations

   Undo adding or subtracting first, then undo multiplying or dividing.

   e.g. $3x + 4 = 19 \Rightarrow 3x = 15 \Rightarrow x = 5$

   $\displaystyle ax + b = c \Rightarrow x = \frac{c - b}{a}$
5. 2.8.5

   Solving Linear Equations With Fractional Coefficients

   Multiply both sides by the reciprocal of the coefficient.

   $\displaystyle \tfrac{2}{3}x = 8 \Rightarrow x = 8 \cdot \tfrac{3}{2} = 12$
6. 2.8.6

   Solving Linear Equations With Decimal Coefficients

   Multiply both sides by 10, 100, … to clear the decimals, then solve.

   $\displaystyle 0.3x + 1.2 = 3 \xrightarrow{\times 10} 3x + 12 = 30$
7. 2.8.7

   Solving Linear Equations With Variables on Both Sides

   Gather the variable terms on one side and the numbers on the other.

   $\displaystyle 5x + 2 = 3x + 10$

   $\displaystyle 2x = 8 \Rightarrow x = 4$
8. 2.8.8

   Solving Linear Equations Containing Binomials

   Expand every product, including a negative sign in front of parentheses. If $x^2$ terms appear on both sides, they cancel and the equation becomes linear. Then collect $x$ terms and solve.

   e.g. $(x+3)(x-2) = (x+1)(x+4)$: $x^2 + x - 6 = x^2 + 5x + 4$, so $-4x = 10$ and $x = -\tfrac52$

9. 2.8.9

   Solving Linear Equations Containing Binomials by Factoring

   When the same binomial appears more than once, treat it as a single unit. Collect it on one side, solve for the binomial, then solve for $x$.

   e.g. $5(x-3) = 2(x-3) + 12 \Rightarrow 3(x-3) = 12 \Rightarrow x - 3 = 4 \Rightarrow x = 7$

   $\displaystyle a(x+h) = b(x+h) + c$

   $\displaystyle \Rightarrow\; (a-b)(x+h) = c$
10. 2.8.10

    Solving Linear Equations by Clearing a Rational Expression

    Multiply every term by the LCD of all the denominators.

    $\displaystyle \frac{x}{3} + \frac{x}{4} = 7 \xrightarrow{\times 12} 4x + 3x = 84$
11. 2.8.11

    Solving Linear Equations Using Cross-Multiplication

    When one fraction equals another, multiply diagonally.

    e.g. $\tfrac{x}{4} = \tfrac{9}{6} \Rightarrow 6x = 36 \Rightarrow x = 6$

    $\displaystyle \frac{a}{b} = \frac{c}{d} \Rightarrow ad = bc$
12. 2.8.12

    Manipulating Expressions and Equations

    Do the same operation to both sides to isolate a variable or to turn one side into the exact expression asked for. On the SAT, often you do not need each variable's value.

    e.g. If $2x + 3 = 7$, then $6x + 9 = 3(2x + 3) = 21$

13. 2.8.13

    Solving Two-Variable Equations

    Solve for one variable in terms of the other, or plug in a given value.

    $\displaystyle 2x + y = 10 \Rightarrow y = 10 - 2x$
14. 2.8.14

    Linear Equations With Infinitely Many Solutions

    The variables cancel and leave something always true. Every number works.

    $\displaystyle 2(x + 1) = 2x + 2 \;\to\; 2 = 2$
15. 2.8.15

    Linear Equations With No Solutions

    The variables cancel and leave something false. No number works.

    $\displaystyle 3x + 1 = 3x + 5 \;\to\; 1 = 5$
16. 2.8.16

    Solving Linear Equations With Unknown Coefficients

    Treat the other letters as numbers and isolate the variable you want. Divide by its coefficient only when that coefficient can’t be zero.

    e.g. Solve $ax - 3 = 7$ for $x$: $x = \tfrac{10}{a}$

    $\displaystyle ax + b = c$

    $\displaystyle \Rightarrow\; x = \frac{c - b}{a} \quad (a \ne 0)$
17. 2.8.17

    Solving Linear Equations With Unknown Coefficients by Factoring

    When the variable appears in more than one term, gather those terms, factor the variable out, then divide.

    e.g. $kx + 2x = 9 \Rightarrow x = \tfrac{9}{k + 2}$

    $\displaystyle ax + bx = c$

    $\displaystyle \Rightarrow\; x(a + b) = c$

    $\displaystyle \Rightarrow\; x = \frac{c}{a + b}$
18. 2.8.18

    Solving Linear Inequalities With Unknown Parameters

    Same steps as an equation. If you divide by a parameter, its sign decides whether the inequality flips.

    $\displaystyle ax > b \;\Rightarrow\; \begin{cases} x > \frac{b}{a} & a > 0 \\[2pt] x < \frac{b}{a} & a < 0 \end{cases}$
19. 2.8.19

    Solving Many-Variable Equations

    To solve a formula for one variable, undo the operations around it until it stands alone.

    e.g. $P = 2l + 2w \Rightarrow w = \tfrac{P - 2l}{2}$

    $\displaystyle A = \tfrac{1}{2}bh \;\Rightarrow\; h = \frac{2A}{b}$

2.9 Modeling With Linear Equations

1. 2.9.1

   Interpreting Linear Expressions

   In $mx + b$, $b$ is the fixed starting amount and $m$ is the amount added for each unit of $x$.

   e.g. $25 + 0.15m$: a $25 fee plus 15¢ per mile

   $\displaystyle \text{total} = \underbrace{m}_{\text{per unit}}\,x + \underbrace{b}_{\text{fixed}}$
2. 2.9.2

   Modeling With Linear Equations

   Name the unknown, write the equation the words describe, solve, and check that the answer makes sense.

   e.g. $30 to join plus $10 a month, $110 spent → $30 + 10m = 110$, $m = 8$ months

3. 2.9.3

   Consecutive Integer Problems

   Call the first integer $n$. Consecutive integers are $n, n+1, n+2$. Consecutive even or odd integers are $n, n+2, n+4$.

   e.g. Three consecutive integers sum to 48 → $3n + 3 = 48$, $n = 15$

   $\displaystyle n + (n + 1) + (n + 2) = 3n + 3$
4. 2.9.4

   Speed, Distance, Time Problems

   Write $d = st$ for each traveler, then connect them: same distance, same time, or distances that add up.

   e.g. Cars leave together at 50 and 60 mph in opposite directions → $110t = 330$ when 330 mi apart, $t = 3$ h

   $\displaystyle d = s\,t$
5. 2.9.5

   Further Speed, Distance, Time Problems

   A chaser closes a gap at the difference of the speeds. Travelers heading toward each other close it at the sum.

   $\displaystyle t = \frac{\text{gap}}{s_1 - s_2} \;\text{(catching up)}$

   $\displaystyle t = \frac{\text{gap}}{s_1 + s_2} \;\text{(approaching)}$
6. 2.9.6

   Modeling Work Problems

   Each worker does a fraction of the job per hour. Working together, the rates add.

   e.g. Pipes fill a tank in 3 h and 6 h → $\tfrac13 + \tfrac16 = \tfrac12$ → together 2 h

   $\displaystyle \frac{1}{a} + \frac{1}{b} = \frac{1}{t}$
7. 2.9.7

   Modeling Mixture Problems

   Track the pure ingredient: concentration times volume, added across the parts.

   e.g. 10 L at 20% plus $x$ L at 50% makes 30% → $2 + 0.5x = 0.3(10 + x)$, $x = 5$

   $\displaystyle c_1V_1 + c_2V_2 = c\,(V_1 + V_2)$

3.10 Graphs of Linear Equations

1. 3.10.1

   The Cartesian Coordinate System

   A point is $(x, y)$: across first, then up. Four quadrants, numbered counterclockwise from upper right.

   e.g. Quadrant I: $(+,+)$, II: $(-,+)$, III: $(-,-)$, IV: $(+,-)$

   $\displaystyle (x, y)$
2. 3.10.2

   Distances Between Points in the Coordinate Plane

   Points that share an $x$ or a $y$: subtract the other coordinates and take the absolute value.

   e.g. $(3, -2)$ to $(3, 5)$ → $7$

   $\displaystyle \text{same } x: \lvert y_2 - y_1 \rvert$

   $\displaystyle \text{same } y: \lvert x_2 - x_1 \rvert$
3. 3.10.3

   Two-Variable Linear Equations and Their Solutions

   A solution is an $(x, y)$ pair that makes the equation true. All solutions together form a line.

   e.g. $(1, 8)$ solves $y = 2x + 6$

4. 3.10.4

   Graphing Linear Equations

   Pick two or three $x$ values, find each $y$, plot the points, draw the line through them.

5. 3.10.5

   Horizontal and Vertical Lines

   $y = b$ is horizontal with slope 0. $x = a$ is vertical with undefined slope.

   $\displaystyle y = b$

   $\displaystyle x = a$
6. 3.10.6

   Calculating Slopes of Straight Lines

   Slope is rise over run between any two points.

   e.g. $(1, 2)$ to $(4, 8)$ → $m = \tfrac63 = 2$

   $\displaystyle m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}}$
7. 3.10.7

   Equations of Lines in Slope-Intercept Form

   $m$ is the slope and $b$ is the $y$-intercept.

   e.g. $y = -2x + 5$: slope $-2$, crosses the $y$-axis at $(0, 5)$

   $\displaystyle y = mx + b$

   $\displaystyle m = \frac{y_2 - y_1}{x_2 - x_1}$
8. 3.10.8

   Properties of Lines Given in Slope-Intercept Form

   Positive $m$ rises, negative $m$ falls, zero is flat. Larger $\lvert m \rvert$ means steeper. The $x$-intercept is where $y = 0$.

   $\displaystyle x\text{-intercept} = -\frac{b}{m}$
9. 3.10.9

   Working With Lines in Slope-Intercept Form

   $m$ is the slope and $b$ is the $y$-intercept. To find the line through two points, compute $m$, then plug in one point to get $b$. Parallel lines share $m$. Perpendicular slopes multiply to $-1$.

   e.g. Through $(1, 5)$ and $(3, 9)$: $m = \tfrac{4}{2} = 2$, $5 = 2(1) + b$, so $y = 2x + 3$

   $\displaystyle y = mx + b$

   $\displaystyle m = \frac{y_2 - y_1}{x_2 - x_1}$
10. 3.10.10

    Equations of Lines in Point-Slope Form

    Use it when you know the slope and one point on the line.

    e.g. Slope 3 through $(2, -1)$: $y + 1 = 3(x - 2)$

    $\displaystyle y - y_1 = m(x - x_1)$
11. 3.10.11

    Equations of Lines in Standard Form

    Integer coefficients with $x$ and $y$ on the same side. The quickest form for finding both intercepts.

    e.g. $2x + 3y = 12$: intercepts $(6, 0)$ and $(0, 4)$

    $\displaystyle Ax + By = C$
12. 3.10.12

    Properties of Lines Given in Standard Form

    Read the slope and intercepts straight from the coefficients.

    $\displaystyle m = -\frac{A}{B}$

    $\displaystyle x\text{-int} = \frac{C}{A} \qquad y\text{-int} = \frac{C}{B}$
13. 3.10.13

    Parallel Lines in the Coordinate Plane

    Parallel lines have the same slope and different $y$-intercepts. All vertical lines are parallel to each other.

    e.g. $y = 2x + 1$ and $y = 2x - 4$ are parallel

    $\displaystyle m_1 = m_2$
14. 3.10.14

    Finding Equations of Parallel Lines

    Keep the slope, then put the given point into point-slope form.

    e.g. Parallel to $y = 3x - 2$ through $(1, 5)$: $y = 3x + 2$

    $\displaystyle y - y_1 = m(x - x_1)$
15. 3.10.15

    Perpendicular Lines in the Coordinate Plane

    Perpendicular slopes are negative reciprocals: they multiply to $-1$. Horizontal and vertical lines are perpendicular.

    e.g. Slope $\tfrac23$ → perpendicular slope $-\tfrac32$

    $\displaystyle m_1 m_2 = -1$

    $\displaystyle m_2 = -\frac{1}{m_1}$
16. 3.10.16

    Finding Equations of Perpendicular Lines

    Flip the slope and change its sign, then use the given point.

    e.g. Perpendicular to $y = 2x + 1$ through $(4, 1)$: $y = -\tfrac12x + 3$

    $\displaystyle y - y_1 = -\frac{1}{m}(x - x_1)$

3.11 Modeling With Two-Variable Linear Equations

1. 3.11.1

   Modeling With Linear Equations in Two Variables

   Find the starting value (the intercept) and the rate (the slope), then write $y = mx + b$.

   e.g. Plant 4 cm tall growing 1.5 cm a week: $h = 1.5w + 4$

2. 3.11.2

   Modeling Totals Using Two-Variable Linear Equations

   Two kinds of item with a fixed total: price times count for each, added.

   e.g. Adult tickets $12, child $8, $200 total: $12a + 8c = 200$

   $\displaystyle ax + by = T$
3. 3.11.3

   Interpreting Coefficients of Two-Variable Equations

   In $y = mx + b$, $m$ is the change in $y$ per one unit of $x$ and $b$ is the value when $x = 0$. In $ax + by = c$, $a$ and $b$ are the per-unit amounts and $c$ is the total.

   e.g. $C = 25 + 0.5m$: a $25 flat fee plus $0.50 per mile

4. 3.11.4

   Analyzing and Interpreting Graphs of Linear Equations

   The slope is the rate of change, in $y$-units per $x$-unit. The $y$-intercept is the value when $x = 0$. The $x$-intercept is where the quantity hits zero.

5. 3.11.5

   Further Modeling With Linear Equations in Two Variables

   From two data points, compute the slope, then use point-slope form.

   e.g. $50 for 2 h, $80 for 4 h → $m = 15$, $C = 15h + 20$

   $\displaystyle m = \frac{\Delta y}{\Delta x}$
6. 3.11.6

   Distance-Time Graphs

   The slope is the speed. A flat stretch means stopped; a steeper line means faster; a falling line means heading back.

   $\displaystyle \text{speed} = \text{slope} = \frac{\Delta d}{\Delta t}$
7. 3.11.7

   Speed-Time Graphs

   The slope is the acceleration. A flat stretch means constant speed, not stopped.

   $\displaystyle \text{acceleration} = \frac{\Delta v}{\Delta t}$
8. 3.11.8

   Calculating Distance From a Speed-Time Graph

   Distance traveled is the area under the graph. Split it into rectangles and triangles.

   e.g. 0 to 10 m/s over 4 s → $\tfrac12 \cdot 4 \cdot 10 = 20$ m

   $\displaystyle d = \text{area under the } v\text{–}t \text{ graph}$

3.12 Systems of Equations

1. 3.12.1

   Introduction to Systems of Linear Equations

   Two equations sharing variables. The solution is where the lines cross: one point, none (parallel), or infinitely many (same line).

2. 3.12.2

   Solving Systems of Equations by Substitution

   Solve one equation for a variable, substitute into the other, solve, then plug back in.

   $\displaystyle y = 2x,\quad x + y = 9$

   $\displaystyle x + 2x = 9 \Rightarrow x = 3,\ y = 6$
3. 3.12.3

   Introduction to the Elimination Method

   Add or subtract the equations so that one variable cancels.

   e.g. So $x = 7$ and $y = 3$

   $\displaystyle \begin{array}{r} x + y = 10 \\ x - y = 4 \\ \hline 2x = 14 \end{array}$
4. 3.12.4

   Solving Systems of Linear Equations Using Elimination: One Transformation

   Multiply one equation so a variable’s coefficients become opposites, then add.

   e.g. $2x + y = 7$, $x - 3y = 7$: triple the first, add → $7x = 28$, $x = 4$, $y = -1$

5. 3.12.5

   Solving Systems of Linear Equations Using Elimination: Two Transformations

   Multiply each equation by the other’s coefficient so one variable matches.

   e.g. $3x + 2y = 16$, $2x - 5y = -2$: ×2 and ×3 → $19y = 38$, $y = 2$, $x = 4$

6. 3.12.6

   Systems of Linear Equations With Fractional Coefficients

   Clear the fractions first: multiply each equation by its LCD.

   $\displaystyle \tfrac{x}{2} + \tfrac{y}{3} = 4 \;\xrightarrow{\times 6}\; 3x + 2y = 24$
7. 3.12.7

   Systems of Linear Equations With Decimal Coefficients

   Multiply by 10 or 100 to turn the decimals into integers.

   $\displaystyle 0.2x + 0.5y = 1.6 \;\xrightarrow{\times 10}\; 2x + 5y = 16$
8. 3.12.8

   Systems of Equations With No Solutions and Infinitely Many Solutions

   Both variables cancel. A false statement like $0 = 5$ means no solution (parallel lines). A true one like $0 = 0$ means infinitely many (the same line).

9. 3.12.9

   Parametrizing Systems With Infinitely Many Solutions

   If both equations describe the same line, every point on it is a solution. Set one variable equal to a parameter $t$ and solve for the other in terms of $t$.

   e.g. $x + 2y = 4$ and $2x + 4y = 8$: let $y = t$, then $x = 4 - 2t$, so the solutions are $(4 - 2t,\ t)$

10. 3.12.10

    Determining Unknown Parameters in Systems of Linear Equations

    Compare the coefficients of $a_1x + b_1y = c_1$ and $a_2x + b_2y = c_2$. Equal slopes with different intercepts means no solution. The same line means infinitely many. Different slopes means exactly one.

    e.g. $3x + ky = 5$ and $6x + 4y = 7$: $\tfrac36 = \tfrac{k}{4}$ gives $k = 2$, and $\tfrac57 \ne \tfrac12$, so no solution when $k = 2$

    $\displaystyle \frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2}:\ \text{none}$

    $\displaystyle \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}:\ \text{infinite}$

    $\displaystyle \frac{a_1}{a_2} \ne \frac{b_1}{b_2}:\ \text{one}$
11. 3.12.11

    Calculating the Intersection of Two Lines

    Set the two expressions for $y$ equal, solve for $x$, then find $y$.

    e.g. $y = 2x + 1$ and $y = -x + 7$ → $3x = 6$ → they meet at $(2, 5)$

    $\displaystyle m_1x + b_1 = m_2x + b_2$
12. 3.12.12

    Solving Systems of Linear Equations by Identifying Structure

    When asked for a combination like $x + y$, add or subtract the equations as given instead of solving for each variable.

    e.g. $3x + y = 10$ and $x + 3y = 6$: adding gives $4x + 4y = 16$, so $x + y = 4$

13. 3.12.13

    Larger Systems of Equations

    With three equations in three unknowns, use pairs of equations to eliminate the same variable twice. Solve the resulting two-variable system, then back-substitute.

    e.g. $x+y+z=6$, $x-y+z=2$, $x+y-z=0$: subtracting the 2nd from the 1st gives $y = 2$, subtracting the 3rd gives $z = 3$, so $x = 1$

14. 3.12.14

    Modeling Using Systems of Linear Equations

    Two unknowns need two equations. Usually one counts the items and one counts their value.

    e.g. 150 tickets, adults $10, children $6, $1,140 → $a + c = 150$, $10a + 6c = 1140$ → $a = 60$

15. 3.12.15

    Modeling Coin Problems Using Systems of Linear Equations

    One equation for how many coins, one for what they’re worth. Work in cents.

    e.g. → $d = 14$ dimes, $n = 16$ nickels

    $\displaystyle n + d = 30$

    $\displaystyle 5n + 10d = 220$

4.13 Linear Inequalities

1. 4.13.1

   Inequalities

   A statement that one side is bigger or smaller. Its solution is a range of numbers.

   $\displaystyle < \;\; > \;\; \le \;\; \ge$
2. 4.13.2

   Compound Inequalities

   “And” means both are true, so $x$ is between. “Or” means at least one is true.

   $\displaystyle -2 < x \le 5$

   $\displaystyle x < -1 \;\text{or}\; x > 4$
3. 4.13.3

   Verifying Solutions of Linear Inequalities

   Substitute the value and check whether the statement is true.

   e.g. Is 3 a solution of $2x - 1 > 4$? $5 > 4$ ✓

4. 4.13.4

   Solving One-Step Linear Inequalities

   Solve like an equation, with one rule: multiplying or dividing by a negative flips the sign.

   $\displaystyle -3x < 12 \Rightarrow x > -4$
5. 4.13.5

   Solving Two-Step Inequalities

   Same order as a two-step equation. Watch for the flip.

   $\displaystyle 5 - 2x \ge 11 \Rightarrow -2x \ge 6 \Rightarrow x \le -3$
6. 4.13.6

   Representing Solutions to Inequalities on Number Lines

   Open circle for < or >. Closed circle for ≤ or ≥. Shade toward the solutions.

7. 4.13.7

   Further Solving Linear Inequalities

   Distribute, collect, move variables to one side; flip if you divide by a negative.

   $\displaystyle 3(x - 2) > 5x + 4$

   $\displaystyle -2x > 10 \Rightarrow x < -5$
8. 4.13.8

   Solving Compound Inequalities

   For $a < bx + c < d$, do each operation to all three parts. Dividing by a negative flips both signs.

   $\displaystyle -1 < 2x + 3 < 9$

   $\displaystyle \Rightarrow\; -4 < 2x < 6$

   $\displaystyle \Rightarrow\; -2 < x < 3$
9. 4.13.9

   Interval Notation

   A square bracket includes the endpoint; a parenthesis excludes it.

   $\displaystyle [a, b]:\ a \le x \le b$

   $\displaystyle (a, b):\ a < x < b$

   $\displaystyle [a, b):\ a \le x < b$
10. 4.13.10

    Unbounded Intervals

    Infinity always takes a parenthesis, because it is not a number you can reach.

    $\displaystyle x \ge 2 \;\to\; [2, \infty)$

    $\displaystyle x < 5 \;\to\; (-\infty, 5)$

    $\displaystyle \text{all reals} \;\to\; (-\infty, \infty)$
11. 4.13.11

    Unions of Intervals

    $\cup$ means “or”: everything in either interval.

    $\displaystyle (-\infty, 1) \cup [4, \infty)$
12. 4.13.12

    Intersections of Intervals

    $\cap$ means “and”: only the part both intervals share.

    $\displaystyle [0, 5] \cap (3, 8] = (3, 5]$
13. 4.13.13

    Compound OR Inequalities

    Solve each part on its own. The answer is the union.

    $\displaystyle x < -2 \;\text{or}\; x \ge 1$

    $\displaystyle \to\; (-\infty, -2) \cup [1, \infty)$
14. 4.13.14

    Optimizing Sums, Differences, Products, and Quotients

    Given a range for each variable, pick the endpoints that push the result furthest. A difference is largest with the biggest first value and the smallest second value. For products and quotients with negatives, test all endpoint pairs.

    e.g. $2 \le x \le 5$, $1 \le y \le 4$: max of $x - y$ is $5 - 1 = 4$, min of $\tfrac{x}{y}$ is $\tfrac24 = \tfrac12$

    $\displaystyle \max(x + y) = x_{\max} + y_{\max}$

    $\displaystyle \max(x - y) = x_{\max} - y_{\min}$

    $\displaystyle \max\tfrac{x}{y} = \tfrac{x_{\max}}{y_{\min}}\ \ (x, y > 0)$

4.14 Two-Variable Linear Inequalities

1. 4.14.1

   Graphing Strict Two-Variable Linear Inequalities

   For < or >, draw the boundary dashed, then shade the side that makes the inequality true. Test $(0, 0)$ if it isn’t on the line.

   $\displaystyle y > mx + b$

   $\displaystyle \to\; \text{dashed, shade above}$
2. 4.14.2

   Graphing Non-Strict Two-Variable Linear Inequalities

   For ≤ or ≥, draw the boundary solid, because points on the line count.

   $\displaystyle y \le mx + b$

   $\displaystyle \to\; \text{solid, shade below}$
3. 4.14.3

   Further Graphing of Two-Variable Linear Inequalities

   Rearrange into $y = \ldots$ first. Dividing by a negative flips the sign, which flips the shading.

   $\displaystyle 2x - 3y > 6 \;\Rightarrow\; y < \tfrac{2}{3}x - 2$
4. 4.14.4

   Systems of Linear Inequalities

   Graph each inequality. The solutions are where all the shadings overlap.

4.15 Modeling With Inequalities

1. 4.15.1

   Introduction to Modeling With Inequalities

   Translate the words: at least ≥, at most ≤, no more than ≤, more than >, fewer than <.

   e.g. “At least 18 years old” → $a \ge 18$

2. 4.15.2

   Modeling With One-Step Inequalities

   Write it, solve it, then round in the direction the situation demands.

   e.g. Bags hold 12 lb, 50 lb of sand → $12b \ge 50$, $b \ge 4.17$ → 5 bags

3. 4.15.3

   Modeling With Two-Step Inequalities

   A fixed amount plus a rate, compared with a limit.

   e.g. $40 budget, $10 entry plus $6 per ride → $10 + 6r \le 40$, $r \le 5$

   $\displaystyle b + rx \le L$
4. 4.15.4

   Modeling With Compound Inequalities

   A quantity that must stay inside a range gives a three-part inequality.

   e.g. Keep milk between 35°F and 40°F → $35 \le T \le 40$

5. 4.15.5

   Modeling With Two-Variable Linear Inequalities

   A budget or capacity limit becomes $ax + by \le C$. Only sensible points count, often whole and nonnegative.

   e.g. $60 for $5 books and $12 games: $5b + 12g \le 60$

6. 4.15.6

   Modeling With Systems of Linear Inequalities

   Each constraint is one inequality. The workable choices lie in the overlap.

7. 4.15.7

   Modeling and Solving Systems of Linear Inequalities Using Technology

   Enter each inequality in a graphing tool, read the overlap, and check its corner points.

5.16 Introduction to Geometry

1. 5.16.1

   Units of Area and Volume

   Area is in square units, volume in cubic units. To convert, square or cube the length conversion.

   $\displaystyle 1\text{ ft}^2 = 12^2 = 144\text{ in}^2$

   $\displaystyle 1\text{ yd}^3 = 3^3 = 27\text{ ft}^3$
2. 5.16.2

   Points, Lines, Rays, and Segments

   A line runs forever both ways. A ray starts at a point and runs one way. A segment has two endpoints.

   $\displaystyle \overleftrightarrow{AB} \qquad \overrightarrow{AB} \qquad \overline{AB}$
3. 5.16.3

   Parallel and Perpendicular Lines

   Parallel lines never meet. Perpendicular lines meet at a right angle.

   $\displaystyle \ell \parallel m \qquad \ell \perp m$
4. 5.16.4

   Measures of Segments

   $AB$ with no bar means the length of $\overline{AB}$, a number.

   $\displaystyle AB = \text{length of } \overline{AB}$
5. 5.16.5

   Congruent Segments

   Same length. Marked with matching tick marks.

   $\displaystyle \overline{AB} \cong \overline{CD} \iff AB = CD$
6. 5.16.6

   Midpoints

   The point that splits a segment into two equal halves.

   $\displaystyle AM = MB = \tfrac{1}{2}AB$
7. 5.16.7

   Collinear Points

   Points that all lie on one line. Any two points are always collinear.

8. 5.16.8

   The Segment Addition Postulate

   If $B$ is on the segment between $A$ and $C$, the pieces add to the whole.

   e.g. $AB = 4$, $BC = 7$ → $AC = 11$

   $\displaystyle AB + BC = AC$

5.17 Angles