Appendices A through O · Glossary · Answer Key Guide · Index · MGU Library Connections
The appendices collect reference material that supports every chapter of this book. Use them when you need a refresher on prerequisite skills, a concise reminder of a rule, or a strategy guide for a common procedure.
This appendix reviews the arithmetic and pre-algebra skills that Algebra I requires. If any section feels unfamiliar or produces consistent errors, return to Arithmetic and Pre-Algebra Foundations in the MGU Library before continuing. These skills are the foundation on which every chapter of this book is built.
The whole numbers are 0, 1, 2, 3, 4, 5, … They have no fractions, no decimals, and no negative values. Place value tells you what each digit represents: ones, tens, hundreds, thousands, and so on. In the number 4,703: the 4 is in the thousands place (= 4,000), the 7 is in the hundreds place (= 700), the 0 is in the tens place, and the 3 is in the ones place. The complete value is 4,000 + 700 + 0 + 3 = 4,703.
Operations review: Addition, subtraction, multiplication, and division of whole numbers should be fluent. Pay attention to any zeros in the middle of a multiplication or long-division problem, as these are common sources of error.
The integers include all whole numbers and their negatives: …, −4, −3, −2, −1, 0, 1, 2, 3, 4, … Positive numbers lie to the right of zero on the number line; negative numbers lie to the left.
| Operation | Rule | Examples |
|---|---|---|
| Addition (same sign) | Add absolute values; keep the sign | −7 + (−4) = −11 |
| Addition (different signs) | Subtract smaller absolute value from larger; keep sign of the larger | 9 + (−3) = 6 −5 + 8 = 3 |
| Subtraction | Rewrite as addition of the opposite: a − b = a + (−b) | 6 − 10 = −4 −3 − (−7) = 4 |
| Multiplication / Division | Same signs → positive. Different signs → negative. | (−3)(−5) = 15 (−4)(6) = −24 |
When an expression contains more than one operation, follow the standard order: (1) Parentheses, (2) Exponents, (3) Multiplication and Division left to right, (4) Addition and Subtraction left to right.
Evaluate: 3 + 4² ÷ 2 − (1 + 5)
= 3 + 4² ÷ 2 − 6 (parentheses first)
= 3 + 16 ÷ 2 − 6 (exponent)
= 3 + 8 − 6 (division)
= 5 (addition and subtraction, left to right)
Multiplication and division have equal priority — do them left to right. The same applies to addition and subtraction. Never multiply before dividing simply because M comes before D in PEMDAS.
| Operation | Procedure | Example |
|---|---|---|
| Equivalent fractions | Multiply or divide numerator and denominator by the same nonzero number | 2/3 = 4/6 = 10/15 |
| Simplifying | Divide numerator and denominator by their GCF | 12/18 = 2/3 |
| Adding/Subtracting | Common denominator required | 1/4 + 2/3 = 3/12 + 8/12 = 11/12 |
| Multiplying | Multiply straight across | 3/5 · 2/7 = 6/35 |
| Dividing | Multiply by the reciprocal of the divisor | 3/4 ÷ 5/6 = 3/4 · 6/5 = 9/10 |
Decimals extend the place-value system to the right of the decimal point: tenths, hundredths, thousandths. A percent is a ratio per 100. To convert a percent to a decimal, divide by 100: 37% = 0.37. To convert a decimal to a percent, multiply by 100: 0.08 = 8%. To find a percent of a number, multiply: 15% of 80 = 0.15 × 80 = 12.
A ratio is a comparison of two quantities by division: a/b or a : b. A proportion is an equation that sets two ratios equal: a/b = c/d. To solve, use cross multiplication: ad = bc, then solve for the unknown.
An exponent tells how many times the base is multiplied by itself: 4³ = 4 · 4 · 4 = 64. Note that (−3)² = 9 (positive) but −3² = −9, because the exponent applies only to 3 without parentheses. Parentheses make all the difference.
A variable is a letter representing a number. An expression combines numbers, variables, and operations with no equals sign. To evaluate, replace each variable with its given value and simplify. Example: evaluate 3x − 2 when x = 5: 3(5) − 2 = 15 − 2 = 13.
| Equation | Inverse Operation | Solution |
|---|---|---|
x + 7 = 12 | Subtract 7 from both sides | x = 5 |
x − 4 = 9 | Add 4 to both sides | x = 13 |
3x = 21 | Divide both sides by 3 | x = 7 |
x/5 = 4 | Multiply both sides by 5 | x = 20 |
The coordinate plane has a horizontal x-axis and a vertical y-axis crossing at the origin (0, 0). Every point is described by an ordered pair (x, y). The axes divide the plane into four quadrants: Quadrant I (upper right, x > 0, y > 0), Quadrant II (upper left), Quadrant III (lower left), and Quadrant IV (lower right, x > 0, y < 0). To plot a point, start at the origin, move horizontally by x, then vertically by y.
These properties hold for all real numbers a, b, and c. When you simplify an expression or solve an equation, you apply one or more of these properties at every step — even when you do not name them.
| Property | Statement | Example |
|---|---|---|
| Commutative | a + b = b + a | 3 + 7 = 7 + 3 |
| Associative | (a + b) + c = a + (b + c) | (2 + 5) + 3 = 2 + (5 + 3) |
| Identity | a + 0 = a | x + 0 = x |
| Inverse | a + (−a) = 0 | 5 + (−5) = 0 |
| Property | Statement | Example |
|---|---|---|
| Commutative | a · b = b · a | 4 · 9 = 9 · 4 |
| Associative | (a · b) · c = a · (b · c) | (2 · 3) · 5 = 2 · (3 · 5) |
| Identity | a · 1 = a | x · 1 = x |
| Inverse | a · (1/a) = 1 (a ≠ 0) | 7 · (1/7) = 1 |
| Zero | a · 0 = 0 | 99 · 0 = 0 |
a(b + c) = ab + ac
a(b − c) = ab − ac
a(b + c + d) = ab + ac + ad
(a + b)(c + d) = ac + ad + bc + bd
| Property | Statement |
|---|---|
| Addition | If a = b, then a + c = b + c |
| Subtraction | If a = b, then a − c = b − c |
| Multiplication | If a = b, then ac = bc |
| Division | If a = b and c ≠ 0, then a/c = b/c |
| Reflexive | a = a |
| Symmetric | If a = b, then b = a |
| Transitive | If a = b and b = c, then a = c |
| Substitution | If a = b, then a may replace b in any expression |
| Operation | Rule |
|---|---|
| Adding | Same signs: add absolute values, keep sign. Different signs: subtract smaller absolute value from larger, keep sign of the larger. |
| Subtracting | Convert to addition: a − b = a + (−b). Then apply addition rules. |
| Multiplying/Dividing | Same signs → positive result. Different signs → negative result. |
−(−a) = a (−1) · a = −a (−a)(−b) = ab
| Operation | Rule |
|---|---|
| Adding/Subtracting | Find LCD. Rewrite both fractions with LCD. Add or subtract numerators; keep denominator. |
| Multiplying | Multiply numerators together; multiply denominators together. Simplify before multiplying when possible. |
| Dividing | Multiply by the reciprocal of the divisor: a/b ÷ c/d = a/b · d/c |
| Mixed numbers | Convert to improper fractions before performing operations. |
The rules for signed numbers apply to fractions. A negative fraction can be written in three equivalent forms: −(a/b) = (−a)/b = a/(−b). The most common convention places the negative sign in front of the fraction.
−(3/4) + (1/2) = −(3/4) + (2/4) = −(1/4)
(−2/3)(−3/5) = 6/15 = 2/5
| Symbol | Name | Meaning | Example |
|---|---|---|---|
+ | Plus | Addition or positive | x + 5 |
− | Minus | Subtraction or negative | x − 3, −7 |
· | Dot | Multiplication | 4 · x |
( ) | Parentheses | Grouping or multiplication | 3(x + 2) |
÷ or / | Division symbol / Slash | Division | x ÷ 4, x/4 |
a/b | Fraction bar | Division | (2x+1)/3 |
aⁿ | Exponent | a multiplied by itself n times | x³ |
√ | Square root | Principal square root | √25 = 5 |
ⁿ√ | nth root | nth root | ³√8 = 2 |
|a| | Absolute value | Distance from zero | |−7| = 7 |
| Symbol | Meaning | Example |
|---|---|---|
= | Equals | 3x = 12 |
≠ | Not equal to | x ≠ 0 |
< | Less than | x < 5 |
> | Greater than | x > −2 |
≤ | Less than or equal to | x ≤ 7 |
≥ | Greater than or equal to | x ≥ 0 |
≈ | Approximately equal to | √2 ≈ 1.414 |
| Symbol | Meaning | Example |
|---|---|---|
x, y, n, t, … | Variable — a letter representing a number | Let x = number of hours |
f(x) | f of x — the output of function f when input is x | f(x) = 3x + 1 |
f(2) | f evaluated at 2 — substitute x = 2 | f(2) = 7 |
(x, y) | Ordered pair — a point on the coordinate plane | (3, −5) |
m | Slope (convention) | Rate of change of a line |
b | y-intercept (convention) | Where a line crosses the y-axis |
| Notation | Meaning | Example |
|---|---|---|
(a, b) | Open interval: a < x < b | (2, 7) means 2 < x < 7 |
[a, b] | Closed interval: a ≤ x ≤ b | [0, 5] means 0 ≤ x ≤ 5 |
[a, b) | Half-open: a ≤ x < b | [1, 4) means 1 ≤ x < 4 |
(−∞, a) | All numbers less than a | x < a |
(a, ∞) | All numbers greater than a | x > a |
ℝ | All real numbers | No restriction on domain |
Use this guide as a step-by-step reference whenever you are solving a linear equation in one variable.
Step 1. Simplify each side independently: distribute to remove parentheses, then combine like terms on each side.
Step 2. Move all variable terms to one side using addition or subtraction.
Step 3. Isolate the variable using multiplication or division to make its coefficient equal to 1.
Step 4. Check: substitute your answer into the original equation and verify both sides are equal.
| Equation Type | Key Move | Example |
|---|---|---|
| One-step | One inverse operation | x + 7 = 12 → subtract 7 |
| Two-step | Undo addition/subtraction first, then multiplication/division | 2x − 3 = 9 |
| With parentheses | Distribute first | 3(x + 4) = 21 |
| Variables on both sides | Move variable terms to one side | 5x = 2x + 9 |
| With fractions | Multiply through by LCD to clear fractions | x/3 + 1 = 5 |
| With decimals | Multiply through by a power of 10 | 0.2x + 1.4 = 3.0 |
| Identity | Simplifies to a statement that is always true | 3x + 6 = 3(x + 2) → all reals |
| No solution | Simplifies to a false statement | 2x + 1 = 2x + 5 → no solution |
When an equation contains fractions, multiply every term on both sides by the LCD to eliminate all fractions before solving.
Solve: x/4 − 1/2 = 3/8
LCD of 4, 2, and 8 is 8. Multiply every term by 8:
8 · (x/4) − 8 · (1/2) = 8 · (3/8)
2x − 4 = 3
2x = 7 → x = 7/2
| Inequality | Meaning | Endpoint | Direction |
|---|---|---|---|
x < a | Less than a | Open circle at a (NOT included) | Arrow to the left |
x > a | Greater than a | Open circle at a (NOT included) | Arrow to the right |
x ≤ a | Less than or equal to a | Closed (filled) circle at a (included) | Arrow to the left |
x ≥ a | Greater than or equal to a | Closed (filled) circle at a (included) | Arrow to the right |
"And" inequality (conjunction). The solution must satisfy both conditions simultaneously. The solution is the overlap (intersection) of the two sets. Example: −2 < x ≤ 5. On the number line, shade the region between −2 (open circle) and 5 (closed circle).
"Or" inequality (disjunction). The solution satisfies at least one condition. The solution is the union of the two sets. Example: x < −1 or x ≥ 4. Shade to the left of −1 (open circle) and to the right of and including 4 (closed circle).
When you multiply or divide both sides of an inequality by a negative number, the inequality symbol reverses direction.
−3x > 12 → x < −4 (symbol reversed)
This rule does NOT apply when adding or subtracting, nor when multiplying or dividing by a positive number.
If f is the name of a function, then f(x) (read "f of x") represents the output when x is the input. This is not multiplication — the parentheses indicate "the function is applied to the input x."
f(x) = 3x − 5
f(2) = 3(2) − 5 = 1
f(0) = 3(0) − 5 = −5
f(−4) = 3(−4) − 5 = −17
The domain is the set of all allowable inputs. The range is the set of all possible outputs. For f(x) = 1/x, the domain excludes 0. For g(x) = √x, the domain is all x ≥ 0.
| Task | Notation | What to Do |
|---|---|---|
| Evaluate at a number | f(3) | Substitute x = 3 |
| Evaluate at an expression | f(a + 1) | Substitute x = a + 1 |
| Find domain | All allowable x | Check for division by zero or even roots of negatives |
| Find range | All possible f(x) values | Consider the behavior of the function across its domain |
All three standard forms below describe the same type of relationship: a line. Each form has its most natural use. Familiarity with all three forms — and the ability to convert between them — is a central Algebra I skill.
y = mx + b
m = slope (rate of change) b = y-intercept (value of y when x = 0)
Best for: graphing a line directly; identifying slope and y-intercept at a glance; writing equations when slope and y-intercept are known.
y − y₁ = m(x − x₁)
m = slope (x₁, y₁) = any known point on the line
Best for: writing an equation when slope and one point are known, or when two points are given.
Ax + By = C
A, B, C are integers; A ≥ 0; A and B are not both zero.
Best for: graphing using intercepts; systems of equations solved by elimination.
x-intercept: set y = 0 and solve. y-intercept: set x = 0 and solve.
| From | To | Method |
|---|---|---|
| Slope-intercept | Standard | Move mx to the left side; multiply through to clear fractions |
| Point-slope | Slope-intercept | Distribute; add y₁ to both sides |
| Standard | Slope-intercept | Solve for y; slope = −A/B |
| Type | Form | Slope | Example |
|---|---|---|---|
| Horizontal | y = k | 0 | y = 3 |
| Vertical | x = k | Undefined | x = −2 |
All variables are assumed to represent nonzero real numbers unless otherwise stated.
| Rule | Statement | Example |
|---|---|---|
| Product Rule | aᵐ · aⁿ = aᵐ⁺ⁿ | x³ · x⁴ = x⁷ |
| Quotient Rule | aᵐ / aⁿ = aᵐ⁻ⁿ | x⁵ / x² = x³ |
| Power Rule | (aᵐ)ⁿ = aᵐⁿ | (x³)⁴ = x¹² |
| Power of a Product | (ab)ⁿ = aⁿbⁿ | (2x)³ = 8x³ |
| Power of a Quotient | (a/b)ⁿ = aⁿ/bⁿ | (x/3)² = x²/9 |
| Zero Exponent | a⁰ = 1 (a ≠ 0) | 7⁰ = 1, x⁰ = 1 |
| Negative Exponent | a⁻ⁿ = 1/aⁿ | x⁻³ = 1/x³ |
| Negative in Denominator | 1/a⁻ⁿ = aⁿ | 1/x⁻⁴ = x⁴ |
A number in scientific notation has the form a × 10ⁿ, where 1 ≤ |a| < 10 and n is an integer.
| Operation | Procedure |
|---|---|
| Multiply | Multiply the decimal factors; add the exponents of 10 |
| Divide | Divide the decimal factors; subtract the exponents of 10 |
| Add/Subtract | Convert to the same power of 10 first, then operate on the decimal parts |
Factoring rewrites a polynomial as a product of simpler expressions. Always check for a GCF first.
6x² − 9x = 3x(2x − 3)
ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y)
| Type | Method | Example |
|---|---|---|
| Leading coefficient 1 | Find p, q such that pq = c and p + q = b: x² + bx + c = (x + p)(x + q) | x² + 7x + 12 = (x + 3)(x + 4) |
| Leading coefficient > 1 | Use the AC method: find factors of ac that add to b; rewrite the middle term; factor by grouping | 2x² + 7x + 3 = (2x + 1)(x + 3) |
Difference of Squares: a² − b² = (a + b)(a − b)
Perfect Square (sum): a² + 2ab + b² = (a + b)²
Perfect Square (diff): a² − 2ab + b² = (a − b)²
Sum of Cubes (Alg II): a³ + b³ = (a + b)(a² − ab + b²)
Diff of Cubes (Alg II): a³ − b³ = (a − b)(a² + ab + b²)
For any quadratic equation ax² + bx + c = 0 where a ≠ 0:
x = (−b ± √(b² − 4ac)) / (2a)
To use the formula: (1) Write the equation in standard form (set equal to zero). (2) Identify a, b, and c. (3) Substitute into the formula. (4) Simplify under the radical sign. (5) Simplify the square root if possible. (6) Evaluate both solutions using + and −.
The discriminant is D = b² − 4ac. Its value predicts the number and type of solutions before you solve.
| Discriminant | Number of Solutions | Type | Graph of y = ax² + bx + c |
|---|---|---|---|
D > 0 | 2 | Two distinct real solutions | Parabola crosses x-axis at two points |
D = 0 | 1 | One real solution (repeated root) | Parabola touches x-axis at exactly one point |
D < 0 | 0 real | Two non-real (complex) solutions | Parabola does not cross the x-axis |
| Form | Equation | Best Used For |
|---|---|---|
| Standard | y = ax² + bx + c | Evaluating; finding y-intercept; using quadratic formula |
| Vertex | y = a(x − h)² + k | Graphing; reading vertex (h, k) directly |
| Factored | y = a(x − p)(x − q) | Reading x-intercepts (zeros) p and q directly |
The x-coordinate of the vertex is x = −b / (2a). Substitute into the equation to find the y-coordinate.
√a = b means b² = a (b ≥ 0)
ⁿ√a = b means bⁿ = a
The principal square root of a is always non-negative: √25 = 5, not −5.
| Rule | Statement | Example |
|---|---|---|
| Product Rule | √(ab) = √a · √b | √12 = √4 · √3 = 2√3 |
| Quotient Rule | √(a/b) = √a / √b | √(9/4) = 3/2 |
| Simplest Radical Form | No perfect-square factors under radical; no fractions under radical; no radical in denominator | √50 = 5√2 |
a^(1/n) = ⁿ√a
a^(m/n) = (ⁿ√a)ᵐ = ⁿ√(aᵐ)
8^(1/3) = ³√8 = 2 27^(2/3) = (³√27)² = 3² = 9
A fraction with a radical in the denominator is not in simplest form. Multiply numerator and denominator by a form of 1 that eliminates the radical.
5/√3 = (5/√3) · (√3/√3) = 5√3/3
| n | n² | n³ |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 4 | 8 |
| 3 | 9 | 27 |
| 4 | 16 | 64 |
| 5 | 25 | 125 |
| 6 | 36 | 216 |
| 7 | 49 | 343 |
| 8 | 64 | 512 |
| 9 | 81 | 729 |
| 10 | 100 | 1,000 |
| 11 | 121 | — |
| 12 | 144 | — |
| 15 | 225 | — |
| 20 | 400 | — |
| Measure | Definition | When to Use |
|---|---|---|
| Mean (average) | Sum of all values ÷ number of values: x̄ = (x₁ + x₂ + … + xₙ) / n | Symmetric data without extreme outliers |
| Median | Middle value when data are ordered. If even number of values, average the two middle values. | Skewed data or data with outliers |
| Mode | The value that appears most frequently. A data set may have one mode, multiple modes, or no mode. | Categorical data; most common value |
| Measure | Formula | Description |
|---|---|---|
| Range | Maximum − Minimum | Distance from smallest to largest value |
| Interquartile Range (IQR) | Q₃ − Q₁ | Spread of the middle 50% of data |
| Mean Absolute Deviation (MAD) | Σ|xᵢ − x̄| / n | Average distance of each value from the mean |
| Display | Best For | Key Features |
|---|---|---|
| Dot plot | Small data sets; seeing individual values | Each dot = one data value on a number line |
| Histogram | Large data sets; distribution shape | Bars show frequency of intervals (bins); no gaps unless no data in interval |
| Box plot | Spread, quartiles, outliers | Box from Q₁ to Q₃; whiskers extend to min and max |
| Scatter plot | Relationship between two variables | Each point represents one pair of values |
| Two-way table | Categorical data with two variables | Rows and columns; cells show counts or percents |
| Correlation | Meaning |
|---|---|
| Positive | As one variable increases, the other tends to increase |
| Negative | As one variable increases, the other tends to decrease |
| No correlation | No consistent pattern in the scatter plot |
Residual = Actual − Predicted
A residual close to zero means the model fits well at that point. A line of best fit minimizes the overall residuals.
Technology is a tool — it does not replace understanding. A calculator gives you an answer, but it cannot tell you whether the problem is set up correctly or what the answer means. Use technology to check, explore, and work efficiently, not to bypass thinking.
| Task | How To | Common Pitfall |
|---|---|---|
| Order of operations | Calculator follows PEMDAS automatically, but parentheses must be entered explicitly | Type (3+4)×2 for 14; 3+4×2 gives 11 |
| Negative numbers | Use the (−) or NEG key, not the subtraction key | −3² and (−3)² give different results; use parentheses |
| Fractions | Use fraction mode or enter as division: 3÷4 for 3/4 | Division order matters: 3÷4 ≠ 4÷3 |
| Square roots | Use the √ key; for higher roots use x^(1/n) or caret ^ | √(9+16) ≠ √9 + √16; use parentheses under the radical |
| Task | Steps |
|---|---|
| Enter a function | Go to Y= editor. Enter function in terms of x. For y = 2x − 3, type 2x − 3 after Y₁=. |
| Set the window | Typical Algebra I settings: x from −10 to 10, y from −10 to 10. Use ZOOM → Standard for default view. |
| Find intercepts | CALC → Zero/Root for x-intercepts; CALC → Value, enter x = 0 for y-intercept. |
| Find intersections | Graph both functions; CALC → Intersect. Useful for solving systems by graphing. |
| Use a table | TABLE feature evaluates a function for multiple x-values at once. |
| Task | Use Technology? | Notes |
|---|---|---|
| Checking arithmetic | Yes | Verify multi-step computations |
| Checking equation solutions | Yes | Substitute and confirm equality |
| Exploring graph behavior | Yes | Compare how changing m or b shifts a line |
| Finding decimal approximations | Yes | √7 ≈ 2.646 |
| Learning factoring | No | Factoring builds algebraic fluency; do by hand |
| Solving one-step equations | No | Algebra builds from these by hand |
| Building modeling intuition | Both | Try by hand first; use technology to confirm |
This checklist covers the core Algebra I skills that Algebra II expects students to have mastered. Use it for self-assessment before transitioning to Algebra II. If you find consistent gaps in any section, return to the corresponding chapters before continuing.
This glossary defines every key term introduced in Algebra I, organized alphabetically. Numbers in parentheses refer to the chapter where the term first appears. For fuller treatment of any term, see the relevant chapter section.
The answer key for all practice sets, word problems, error analysis problems, and chapter checkpoints is provided in the separate Algebra I Answer Key and Solution Manual, available in the MGU Library. The answer key for each chapter is organized as follows.
Instructors may request the Algebra I Instructor Edition, which includes additional worked examples, suggested pacing guides, alternative problem sets by difficulty, assessment rubrics for written responses, and a chapter-by-chapter prerequisite skill map. Contact the MGU Library for access.
Numbers refer to chapter and section (e.g., 3.7 = Chapter 3, Section 7). Where a concept appears across multiple sections, all locations are listed.
The following entries in the MGU Library reference system are connected to this textbook. These links represent the broader MGU curriculum in which Algebra I serves as a foundation.
Algebra I follows Arithmetic and Pre-Algebra Foundations and precedes Algebra II, Geometry, Trigonometry and Precalculus, Calculus I, Discrete Mathematics, Calculus II, Linear Algebra, Calculus III, Probability and Statistics, and Differential Equations. For the Library edition, Algebra I is the bridge from arithmetic into symbolic reasoning and functions. Together with Algebra II, it makes students fluent in the language of modern mathematics before they enter geometry, trigonometry, calculus, statistics, physics, economics, engineering, computer science, and finance.