Unit 1
Real Numbers & Operations
The real number system includes natural numbers, integers, rationals, and irrationals. Understanding number properties is foundational for all algebra.
🔑 Key Properties to Memorize
Commutative: \(a+b=b+a\) and \(ab=ba\)
Associative: \((a+b)+c=a+(b+c)\)
Distributive: \(a(b+c)=ab+ac\)
Identity: \(a+0=a\) and \(a\cdot1=a\)
Inverse: \(a+(-a)=0\) and \(a\cdot\tfrac{1}{a}=1\)
📝 Example
Simplify: \(-3(x-4)+2x\)\(= -3x+12+2x = -x+12\)
✓ Answer: \(-x+12\)
Unit 2
Solving Linear Equations & Inequalities
Use inverse operations to isolate the variable. For inequalities, flip the inequality sign when multiplying or dividing by a negative number.
Linear Equation: \(ax + b = c \;\Rightarrow\; x = \dfrac{c-b}{a}\)
🔑 Special Cases
No Solution: Variables cancel, false statement (e.g., \(3=5\))
All Real #s: Variables cancel, true statement (e.g., \(3=3\))
Inequality: \(-2x>6 \Rightarrow x<-3\) (flip sign!)
📝 Example
Solve: \(3(x-2)=2x+1\)\(3x-6=2x+1 \Rightarrow x=7\)
✓ Answer: \(x=7\)
Unit 3
Graphing & Writing Linear Equations
A linear equation in two variables produces a straight line. Key forms allow us to read slope and intercepts directly.
🔑 Forms of Linear Equations
Slope-Int: \(y = mx + b\) — slope \(m\), y-int \(b\)
Std. Form: \(Ax + By = C\)
Point-Slope: \(y - y_1 = m(x - x_1)\)
Slope: \(m = \dfrac{y_2 - y_1}{x_2 - x_1}\)
Parallel: same slope; Perpendicular: slopes are negative reciprocals
📝 Example
Find the slope of the line through \((2, 5)\) and \((6, 13)\).\(m = \dfrac{13-5}{6-2} = \dfrac{8}{4} = 2\)
✓ Answer: slope \(= 2\)
Unit 4
Systems of Equations
A system of two linear equations can be solved by substitution, elimination, or graphing. The solution is the ordered pair \((x,y)\) that satisfies both equations.
🔑 Methods & Solution Types
Substitution: Isolate one variable, substitute into other equation
Elimination: Add/subtract equations to cancel a variable
One Solution: Lines intersect — different slopes
No Solution: Parallel lines — same slope, different y-int
Infinite: Same line — identical equations
📝 Example
Solve: \(x + y = 7\) and \(x - y = 1\)Add: \(2x=8\Rightarrow x=4\). Then \(y=3\).
✓ Answer: \((4, 3)\)
Unit 5
Exponents & Polynomials
Exponent rules govern how powers behave. Polynomials are expressions with multiple terms combined using addition and subtraction.
🔑 Exponent Rules
Product: \(x^a \cdot x^b = x^{a+b}\)
Quotient: \(\dfrac{x^a}{x^b} = x^{a-b}\)
Power: \((x^a)^b = x^{ab}\)
Zero: \(x^0 = 1\ (x\neq0)\)
Negative: \(x^{-n} = \dfrac{1}{x^n}\)
📝 Example
Expand: \((2x+3)(x-5)\)\(= 2x^2-10x+3x-15 = 2x^2-7x-15\)
✓ Answer: \(2x^2-7x-15\)
Unit 6
Factoring Polynomials
Factoring reverses the multiplication process. It is essential for solving quadratic equations and simplifying expressions.
🔑 Factoring Techniques
GCF: Factor out the greatest common factor first
Trinomial: \(x^2+bx+c = (x+p)(x+q)\) where \(p+q=b,\ pq=c\)
Diff. of Sq: \(a^2-b^2 = (a+b)(a-b)\)
Perfect Sq: \(a^2\pm2ab+b^2 = (a\pm b)^2\)
📝 Example
Factor: \(x^2-5x-14\)Need \(p+q=-5\) and \(pq=-14\): use \(-7\) and \(2\).
✓ Answer: \((x-7)(x+2)\)
Unit 7
Quadratic Equations & Functions
A quadratic function \(f(x)=ax^2+bx+c\) produces a parabola. Solve quadratic equations by factoring, completing the square, or the quadratic formula.
Quadratic Formula: \(\displaystyle x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
🔑 Vertex & Discriminant
Vertex x: \(x = -\dfrac{b}{2a}\)
Discriminant: \(\Delta = b^2-4ac\)
\(\Delta > 0\): Two distinct real roots
\(\Delta = 0\): One repeated real root
\(\Delta < 0\): No real roots (complex)
📝 Example
Solve: \(x^2-5x+6=0\)Factor: \((x-2)(x-3)=0\)
✓ Answer: \(x=2\) or \(x=3\)
Practice Exam
Choose the best answer for each question. Immediate feedback will appear after each selection.
Q 01
Unit 1 · Real Numbers
Which property is illustrated by the statement \(5 \cdot (x + 3) = 5x + 15\)?
(A) Commutative Property of Multiplication
(B) Associative Property of Addition
(C) Identity Property of Multiplication
(D) Distributive Property
(E) Associative Property of Multiplication
Q 02
Unit 1 · Real Numbers
Simplify: \(-4(2x - 3) + 5x\)
(A) \(-3x - 12\)
(B) \(-3x + 12\)
(C) \(3x + 12\)
(D) \(3x - 12\)
(E) \(-13x + 12\)
Q 03
Unit 2 · Linear Equations
Solve for \(x\): \(2(x + 4) = 3x - 1\)
(A) \(x = 7\)
(B) \(x = -7\)
(C) \(x = 9\)
(D) \(x = -9\)
(E) \(x = 3\)
Q 04
Unit 2 · Inequalities
Solve the inequality: \(-3x + 5 \leq 14\). Which is the correct solution?
(A) \(x \leq -3\)
(B) \(x \leq 3\)
(C) \(x \geq 3\)
(D) \(x \leq -3\)
(E) \(x \geq -3\)
Q 05
Unit 3 · Linear Equations
What is the slope and y-intercept of the line \(3x - 4y = 8\)?
(A) slope = 3, y-intercept = -8
(B) slope = 3/4, y-intercept = -2
(C) slope = -3/4, y-intercept = 2
(D) slope = 4/3, y-intercept = -2
(E) slope = 3, y-intercept = 8
Q 06
Unit 3 · Linear Equations
Which equation represents the line passing through \((1, 2)\) and \((3, 8)\)?
(A) y = 2x + 1
(B) y = 3x − 1
(C) y = 3x + 1
(D) y = 2x + 3
(E) y = 6x − 4
Q 07
Unit 3 · Parallel Lines
Line \(\ell\) has equation \(y = 2x + 5\). Which line is perpendicular to \(\ell\)?
(A) y = 2x − 3
(B) y = −2x + 1
(C) y = ½x + 4
(D) y = −½x + 3
(E) y = 2x + 3
Q 08
Unit 4 · Systems
Solve the system: \(2x + y = 10\) and \(x - y = 2\).
(A) (2, 6)
(B) (3, 5)
(C) (4, 2)
(D) (6, -2)
(E) (5, 0)
Q 09
Unit 4 · Systems
Which system has no solution?
(A) y = 2x + 1 and y = −2x + 1
(B) y = 3x − 4 and y = 3x − 4
(C) y = x + 2 and y = −x + 4
(D) y = 2x + 3 and y = 2x − 5
(E) y = 4x and y = x + 6
Q 10
Unit 5 · Exponents
Simplify: \(\dfrac{x^6 \cdot x^2}{x^4}\)
(A) x¹²
(B) x⁶
(C) x⁴
(D) x³
(E) x²
Q 11
Unit 5 · Polynomials
Expand and simplify: \((x + 5)(x - 5)\)
(A) x² + 25
(B) x² − 25
(C) x² − 10x − 25
(D) x² + 10x + 25
(E) x² − 10x + 25
Q 12
Unit 5 · Polynomials
Which expression equals \((3x - 2)^2\)?
(A) 9x² + 4
(B) 9x² − 4
(C) 9x² + 12x + 4
(D) 9x² − 12x + 4
(E) 3x² − 12x + 4
Q 13
Unit 6 · Factoring
Factor completely: \(x^2 + 7x + 12\)
(A) (x + 3)(x + 4)
(B) (x + 2)(x + 6)
(C) (x − 3)(x − 4)
(D) (x + 1)(x + 12)
(E) (x − 3)(x + 4)
Q 14
Unit 6 · Factoring
Factor: \(4x^2 - 36\)
(A) (2x − 6)²
(B) 4(x² − 9)
(C) (2x − 6)(2x + 6)
(D) 2(2x² − 18)
(E) 4(x − 3)(x + 3)
Q 15
Unit 7 · Quadratics
Solve: \(x^2 - 7x + 10 = 0\)
(A) x = 2 or x = −5
(B) x = −2 or x = −5
(C) x = 2 or x = 5
(D) x = −2 or x = 5
(E) x = 7 or x = 10
Q 16
Unit 7 · Quadratics
Use the quadratic formula to solve \(2x^2 + 5x - 3 = 0\). What are the solutions?
(A) x = 1/2 or x = −3
(B) x = −1/2 or x = 3
(C) x = 1 or x = −3/2
(D) x = −1 or x = 3/2
(E) x = 2 or x = −3
Q 17
Unit 7 · Quadratics
The vertex of the parabola \(y = x^2 - 6x + 11\) is at which point?
(A) (3, −2)
(B) (−3, 2)
(C) (3, 2)
(D) (6, 11)
(E) (−6, 11)
Q 18
Unit 7 · Discriminant
How many real solutions does \(x^2 - 4x + 7 = 0\) have?
(A) Two distinct real solutions
(B) Exactly one real solution (repeated)
(C) Infinitely many real solutions
(D) No real solutions
(E) Exactly three real solutions
Q 19
Unit 2 · Word Problem
A train travels at a speed of \((3x + 10)\) mph for \(2\) hours. If the total distance is 80 miles, what is \(x\)?
(A) x = 5
(B) x = 10
(C) x = 15
(D) x = 20
(E) x = 30
Q 20
Unit 4 · Word Problem
A jar has nickels and dimes totaling 18 coins worth $1.35. How many dimes are there?
(A) 5
(B) 7
(C) 9
(D) 11
(E) 13
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Answer Key & Solutions
Full explanations for all 20 questions