Algebra 1 Ultimate Practice Exam

UNIT 01Real Numbers & Properties

The four key number properties: Commutative (a+b=b+a), Associative ((a+b)+c=a+(b+c)), Distributive (a(b+c)=ab+ac), Identity (a+0=a, a×1=a).

a(b + c) = ab + ac | a(b - c) = ab - ac

EXSimplify 2(3x+4) = 6x + 8

UNIT 02Solving Linear Equations

Use inverse operations to isolate the variable. Always perform the same operation on both sides. Check by substituting back.

ax + b = c → x = (c - b) / a

EXSolve 3x + 5 = 20: subtract 5 → 3x = 15 → divide by 3 → x = 5

UNIT 03Inequalities

Solve like equations, but flip the inequality sign when multiplying or dividing by a negative number. Use open circle for < or >, closed circle for ≤ or ≥.

-ax > b → x < -b/a (sign flips!)

EXSolve -3x > 9: divide by -3 (flip!) → x < -3

UNIT 04Functions & Relations

A function assigns exactly one output (y) for every input (x). Test: no x-value repeats in a set of ordered pairs (vertical line test on graphs).

f(x) notation: substitute x into the rule to find f(x)

EXf(x) = 2x − 1 → f(3) = 2(3) − 1 = 5

UNIT 05Linear Functions & Slope

Slope = rise ÷ run. Slope-intercept form: y = mx + b where m is slope and b is y-intercept. Parallel lines have equal slopes; perpendicular lines have negative reciprocal slopes.

m = (y₂ − y₁) / (x₂ − x₁) | y = mx + b

EXSlope between (1,2) and (3,8): m = (8−2)/(3−1) = 6/2 = 3

UNIT 06Linear Equations (Forms)

Point-slope: y − y₁ = m(x − x₁). Standard form: Ax + By = C (A, B, C are integers, A ≥ 0). Convert between forms by algebraic manipulation.

Point-slope: y − y₁ = m(x − x₁)

EXThrough (2,3) slope=4: y − 3 = 4(x − 2) → y = 4x − 5

UNIT 07Systems of Equations

Three methods: Graphing (find intersection), Substitution (solve one for a variable, plug in), Elimination (add/subtract equations to cancel a variable).

Substitution: solve for y, then substitute into other equation

EXy = x + 2 and y = 3x → x + 2 = 3x → x = 1, y = 3 → Solution: (1,3)

UNIT 08Exponents

Key rules: Product: xᵃ·xᵇ = xᵃ⁺ᵇ | Quotient: xᵃ/xᵇ = xᵃ⁻ᵇ | Power: (xᵃ)ᵇ = xᵃᵇ | Zero: x⁰ = 1 (x≠0)

xᵃ · xᵇ = xᵃ⁺ᵇ | (xᵃ)ᵇ = xᵃᵇ | (xy)ⁿ = xⁿyⁿ

EX(x³)² = x⁶ | x⁵/x² = x³ | (2x²)(3x³) = 6x⁵

UNIT 09Polynomials & Factoring

To factor x² + bx + c, find two numbers that multiply to c and add to b. Perfect square: (a+b)² = a² + 2ab + b². Difference of squares: a² − b² = (a+b)(a−b).

x² + bx + c = (x + p)(x + q) where p·q = c and p+q = b

EXFactor x² + 7x + 12: need p·q=12 and p+q=7 → p=3, q=4 → (x+3)(x+4)

UNIT 10Quadratic Equations

Solve by factoring (set each factor = 0) or the quadratic formula. The discriminant b²−4ac tells you: positive = 2 real solutions, zero = 1 solution, negative = no real solutions.

x = (−b ± √(b²−4ac)) / (2a) for ax²+bx+c = 0

EXSolve x²−x−6=0: factor (x−3)(x+2)=0 → x=3 or x=−2

Algebra 1

Concept Review