Math Review Β· Elementary Algebra

π‘₯ Algebra Essentials

Real numbers, exponents, polynomials, equations, inequalities, graphing, linear systems, and 30+ word problems from the Math first and second year course review.

Elementary algebra lessons and practice topics

Number Classifications (#1)

  • Natural numbers β„•: 1, 2, 3, … (counting numbers)
  • Whole numbers π•Ž: 0, 1, 2, … (naturals + zero)
  • Integers β„€: …, βˆ’2, βˆ’1, 0, 1, 2, … (all whole numbers, positive and negative)
  • Rational numbers β„š: any number of the form \(\frac{p}{q}\) with integers \(p, q\) and \(q \neq 0\). Includes terminating and repeating decimals.
  • Irrational numbers: non-terminating, non-repeating decimals β€” e.g., \(\sqrt{2}\), \(\pi\), \(\frac{22}{7}\approx\pi\) (but \(\frac{22}{7}\) is rational).
  • Real numbers ℝ: all rational and irrational numbers combined.

Given set: \(\{-13.3,\ -\frac{1}{7},\ -2,\ -\frac{5}{8},\ 0,\ 3,\ \frac{22}{7},\ \sqrt{5},\ \frac{14}{7}\}\)
Natural: 3, \(\frac{14}{7}=2\)  |  Whole: 0, 2, 3  |  Integers: βˆ’2, 0, 2, 3  |  Rational: βˆ’13.3, \(-\frac{1}{7}\), βˆ’2, \(-\frac{5}{8}\), 0, 3, \(\frac{22}{7}\), 2  |  Irrational: \(\sqrt{5}\)

Absolute Value & Comparisons (#3)

\(|x|\) = distance from zero β€” always non-negative. \(|-7| = 7\), \(|5| = 5\).

\(|-7| > |5|\)  because \(7 > 5\)  |  \(|-15| - |-15| = 15 - 15 = 0\)

Order of Operations β€” PEMDAS (#4)

Parentheses → Exponents → Multiply/Divide (left→right) → Add/Subtract (left→right)

Worked Example #4a β€” Evaluate \(8 \div 2 \cdot 4 - 7 + 3\)

1
Left to right Γ· and Γ—: \(8 \div 2 = 4\), then \(4 \cdot 4 = 16\)
2
Left to right + and βˆ’: \(16 - 7 = 9\), then \(9 + 3 = \mathbf{12}\)

Worked Example #4b β€” Evaluate \(5 + 3(a+b)^2 - a + 3\) when \(a=2, b=-6\)

1
Inside parentheses: \(a+b = 2+(-6) = -4\)
2
Exponent: \((-4)^2 = 16\)
3
Multiply: \(3(16) = 48\)
4
\(5 + 48 - 2 + 3 = \mathbf{54}\)

Integer Exponent Rules (#5)

  • Zero exponent: \(a^0 = 1\) for \(a \neq 0\). Note: \(-7^0 = -(7^0) = -1\) (exponent applies to 7 only)
  • Negative exponent: \(a^{-n} = \dfrac{1}{a^n}\)   e.g., \(2^{-1} = \dfrac{1}{2}\)
  • Product rule: \(a^m \cdot a^n = a^{m+n}\)   e.g., \(-x^3 \cdot x^5 = -x^8\)
  • Quotient rule: \(\dfrac{a^m}{a^n} = a^{m-n}\)
  • Power rule: \((a^m)^n = a^{mn}\)   e.g., \((x^3y^4)^2 = x^6y^8\)
  • Power of product: \((ab)^n = a^nb^n\)
  • Power of quotient: \(\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}\)
Leave all answers with positive exponents.

Worked Example β€” Simplify \((3a^{-1}b^3)^{-2}\)

1
Apply power: \(3^{-2} \cdot a^{2} \cdot b^{-6}\)
2
Positive exponents: \(\dfrac{a^2}{9b^6}\)

Worked Example β€” Simplify \((2x^{-3})(3x^4y^{-1})^2\)

1
Expand power: \((3x^4y^{-1})^2 = 9x^8y^{-2}\)
2
Multiply: \(2x^{-3} \cdot 9x^8y^{-2} = 18x^5y^{-2}\)
3
Positive exponents: \(\dfrac{18x^5}{y^2}\)

Practice Problems β€” MTH 098 #5

a) \(2^{-1}\)

\(\dfrac{1}{2}\)

b) \(-7^0\)

\(-1\). The exponent applies to 7 only, not the minus sign.

c) \(-x^3 \cdot x^5\)

\(-x^8\)

d) \((x^3y^4)^2\)

\(x^6y^8\)

e) \((3a^{-1}b^3)^{-2}\)

\(\dfrac{a^2}{9b^6}\)

f) \(\dfrac{-3a^5b^3}{7a^3b^4}\)

\(\dfrac{-3a^2}{7b}\)

g) \(\dfrac{20a^8b^{11}}{16a^{12}b^9}\)

\(\dfrac{5b^2}{4a^4}\)

Polynomial Types & Degree (#6)

  • Monomial: one term β€” e.g., \(7a^2b^6c\) (degree = 2+6+1 = 9)
  • Binomial: two terms β€” e.g., \(4x^2 - 5y\)
  • Trinomial: three terms β€” e.g., \(3x^3 + 2x^5 - 6\) (degree = 5)

Special Multiplication Patterns (#7)

\((a+b)(a-b) = a^2 - b^2 \quad \text{(Difference of Squares)}\)
\((a+b)^2 = a^2 + 2ab + b^2 \qquad (a-b)^2 = a^2 - 2ab + b^2\)

Worked Example #7g β€” Expand \((x+7)(x-7)\)

1
Difference of squares: \((a+b)(a-b) = a^2-b^2\) with \(a=x, b=7\)
2
Result: \(x^2 - 49\)

Worked Example #7i β€” Expand \((3x-4)^2\)

1
Pattern: \(a^2 - 2ab + b^2\) with \(a=3x, b=4\)
2
\(9x^2 - 24x + 16\)

Practice β€” Selected #7 Problems

\((5x+3)(2x+7)\)

FOIL: \(10x^2 + 35x + 6x + 21 = 10x^2 + 41x + 21\)

\((4y-3)(4y+3)\)

Difference of squares: \(16y^2 - 9\)

\((x+5y)^2\)

\(x^2 + 10xy + 25y^2\)

Solving Linear Equations (#9)

Goal: isolate the variable. Apply the same operation to both sides.

Addition Property: if \(a=b\) then \(a+c=b+c\)
Multiplication Property: if \(a=b\) then \(ac=bc\)

Worked Example #9a β€” Solve \(3a + 5 = 5a + 7\)

1
Move variables: \(3a - 5a = 7 - 5\)
2
\(-2a = 2\) β†’ \(a = -1\)

Solving Formulas for a Variable (#10)

Treat the target variable like you'd isolate \(x\) β€” move everything else to the other side.

Solve \(A = \frac{1}{2}bh\) for \(h\).

Multiply both sides by 2, divide by \(b\): \(h = \dfrac{2A}{b}\)

Solve \(P = 2L + 2W\) for \(W\).

\(P - 2L = 2W\) β†’ \(W = \dfrac{P-2L}{2}\)

Solving Quadratic Equations (#12)

  • Factoring: set each factor equal to zero
  • Square root: if \(x^2 = k\) then \(x = \pm\sqrt{k}\)
  • Quadratic formula: \(x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

Worked Example #12c β€” Solve \(x^2 + 10x + 24 = 0\)

1
Find two numbers: product = 24, sum = 10 β†’ 4 and 6
2
Factor: \((x+4)(x+6) = 0\)
3
Solutions: \(x = -4\) or \(x = -6\)

Practice β€” Selected #9 & #12

#9b: \(2(3-4x)+4 = 2-3(4-4x)\)

Distribute: \(6-8x+4 = 2-12+12x\) β†’ \(10-8x = -10+12x\) β†’ \(20=20x\) β†’ \(x=1\)

#12a: \(3x^2 + 5x = 0\)

Factor: \(x(3x+5)=0\) β†’ \(x=0\) or \(x=-\dfrac{5}{3}\)

#12b: \(4x^2 = 25\)

\(x^2 = \dfrac{25}{4}\) β†’ \(x = \pm\dfrac{5}{2}\)

#12d: \(3x^2 = 21x - 18\)

\(3x^2-21x+18=0\) β†’ \(x^2-7x+6=0\) β†’ \((x-1)(x-6)=0\) β†’ \(x=1\) or \(x=6\)

Solving Linear Inequalities (#13)

Same rules as equations with one critical difference: flip the inequality sign when multiplying or dividing by a negative number.

Express solutions in: set-builder \(\{x \mid x \leq -2\}\) and interval notation \((-\infty, -2]\)

Use [ or ] for ≀ or β‰₯ (endpoint included). Use ( or ) for < or > (endpoint excluded). Always use ( ) with ∞.

Worked Example #13b β€” Solve \(-4y \leq 16\)

1
Divide by \(-4\) β€” flip the sign!
2
\(y \geq -4\)   β†’   \([-4, \infty)\)

Worked Example #13c β€” Solve \(4x + 19 \leq 11\)

1
\(4x \leq 11 - 19 = -8\)
2
\(x \leq -2\)   β†’   \((-\infty, -2]\)

Compound Inequalities (#14)

  • "And" / between (intersection ∩): \(-2 \leq 3x+7 \leq 4\) β€” solve all three parts simultaneously.
  • "Or" (union βˆͺ): \(x \leq 0\) or \(x \geq 6\) β€” each side separately.

Worked Example #14a β€” Solve \(-2 \leq 3x + 7 \leq 4\)

1
Subtract 7 from all parts: \(-9 \leq 3x \leq -3\)
2
Divide by 3: \(-3 \leq x \leq -1\)   β†’   \([-3, -1]\)

Practice β€” #13 & #14

#13a: \(x + 2 \leq 5\)

\(x \leq 3\)   β†’   \((-\infty, 3]\)

#13d: \(4(2x-5) \leq 12x+4\)

\(8x-20\leq 12x+4\) β†’ \(-24\leq 4x\) β†’ \(x\geq -6\)   β†’   \([-6,\infty)\)

#14b: \(3 \leq 2x-1 \leq 9\)

Add 1: \(4 \leq 2x \leq 10\) β†’ divide by 2: \(2 \leq x \leq 5\)   β†’   \([2, 5]\)

#14d: \(9 \leq 3-2x \leq 12\)

Subtract 3: \(6 \leq -2x \leq 9\). Divide by \(-2\), flip both signs: \(-\dfrac{9}{2} \leq x \leq -3\)   β†’   \([-4.5, -3]\)

#14e (or): \(x \leq 0\) or \(x \geq 6\)

Union: \((-\infty, 0] \cup [6, \infty)\)

Slope-Intercept Form (#16–18)

\(y = mx + b\)   where \(m\) = slope, \(b\) = y-intercept

Slope: \(m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{\text{rise}}{\text{run}}\)

Parallel & Perpendicular Lines

  • Parallel: equal slopes β€” \(m_1 = m_2\)
  • Perpendicular: negative reciprocal slopes β€” \(m_1 \cdot m_2 = -1\)

Worked Example #16a β€” Line through \((-2,2)\) and \((1,3)\)

1
Slope: \(m = \dfrac{3-2}{1-(-2)} = \dfrac{1}{3}\)
2
Point-slope: \(y - 2 = \dfrac{1}{3}(x + 2)\)
3
Slope-intercept: \(y = \dfrac{1}{3}x + \dfrac{8}{3}\)

Practice

#16b: Line through \((3,1)\) perpendicular to \(3x-5y=6\).

Solve for \(y\): \(y=\frac{3}{5}x-\frac{6}{5}\). Slope = \(\frac{3}{5}\). Perpendicular slope = \(-\frac{5}{3}\).
\(y-1=-\frac{5}{3}(x-3)\) β†’ \(y=-\frac{5}{3}x+6\)

#17a: Slope \(-\frac{2}{5}\), y-intercept \((0,4)\).

\(y = -\dfrac{2}{5}x + 4\)

#47a: Model \(y = -7.5x + 435\). Find \(x\) when \(y = 390\).

\(-7.5x + 435 = 390\) β†’ \(-7.5x = -45\) β†’ \(x = 6\). In year 2006, about 390 drive-in theaters.

Solving Systems of Linear Equations (#20–21)

  • Graphing: find the intersection point visually
  • Substitution: solve one equation for a variable; plug into the other
  • Elimination (Addition): multiply equations to match a coefficient, then add to cancel one variable

Worked Example #21b β€” \(y = 3x-4\) and \(y = -2x+11\)

1
Set equal: \(3x-4 = -2x+11\)
2
\(5x = 15\) β†’ \(x = 3\)
3
\(y = 3(3)-4 = 5\). Solution: \((3,\ 5)\)

Practice

#21a: \(5x+5y=5\) and \(x=3-y\)

Sub \(x=3-y\): \(5(3-y)+5y=5\) β†’ \(15=5\). No unique solution β€” infinitely many solutions (dependent system).

#21c: \(4x-7y=-19\) and \(6x+5y=18\)

Multiply eq.1 by 3, eq.2 by 2: \(12x-21y=-57\) and \(12x+10y=36\). Subtract: \(-31y=-93\) β†’ \(y=3\). Then \(x=\frac{1}{2}\). Solution: \(\left(\frac{1}{2}, 3\right)\)

#21d: \(2x-3y=14\) and \(4x-6y=28\)

Eq.2 is exactly 2Γ—Eq.1 β€” dependent system. Infinitely many solutions.

Word Problem Strategy

  • Define your variable(s) clearly with units
  • Write the equation(s)
  • Solve algebraically
  • Answer in a full sentence with units

Problem #22 β€” Perimeter

Intro

A rectangular building has a perimeter of 104 feet and a width of 17 feet. Find the length.

Let \(L\) = length.  \(P = 2L + 2W\)
\(104 = 2L + 2(17)\) β†’ \(104 = 2L + 34\) β†’ \(2L = 70\) β†’ \(\mathbf{L = 35 \text{ ft}}\)

Problem #23 β€” Circumference

Intro

A plane circles Rochester with radius 1.5 miles. How far per loop? (Nearest tenth.)

\(C = 2\pi r = 2\pi(1.5) \approx \mathbf{9.4 \text{ miles}}\)

Problem #24 β€” Isosceles Triangle

Intro

An isosceles triangle has two equal sides. The third side is 8 inches less than one equal side. Perimeter = 46 in. Find all three sides.

Let \(s\) = equal side. Third side = \(s-8\).
\(s + s + (s-8) = 46\) β†’ \(3s = 54\) β†’ \(s = 18\).
18 in, 18 in, 10 in

Problem #25 β€” Investment

Mid

$10,000 invested at 5% and 3.5% simple interest. Total interest after one year = $410. How much in each account?

Let \(x\) = amount at 5%.
\(0.05x + 0.035(10000-x) = 410\)
\(0.015x = 60\) β†’ \(x = 4000\)
$4,000 at 5% and $6,000 at 3.5%

Problem #28 β€” Distance/Rate/Time

Mid

Two trains leave the same station in opposite directions. Train A: 85 mph east. Train B: 65 mph west. How long until 540 miles apart?

Combined speed = \(85+65 = 150\) mph.
\(150t = 540\) β†’ \(\mathbf{t = 3.6 \text{ hours}}\) (3 hr 36 min)

Problem #29 β€” Catch-Up Rate

Mid

Dad walks 3 mph and leaves 20 minutes early. You walk 2 mph faster. How long until you overtake him? How far?

Your speed = 5 mph. Dad's head start = \(3 \times \frac{20}{60} = 1\) mile.
\(5t = 3t+1\) β†’ \(t = 0.5\) hr. Distance = \(5(0.5) = \mathbf{2.5 \text{ miles in 30 minutes}}\)

Problem #35 β€” Cell Phone Minutes

Mid

A cell plan: $0.99 flat for 20 minutes, then $0.15/extra minute. How many minutes for $5.00?

\(0.99 + 0.15m = 5.00\) β†’ \(0.15m = 4.01\) β†’ \(m \approx 26\) extra minutes. Total: 46 minutes

Problem #40 β€” Coins

Mid

Emily has 40 coins (dimes and quarters) worth $7.30 total. How many of each?

\(d+q=40\) and \(0.10d+0.25q=7.30\). Sub \(d=40-q\):
\(4-0.10q+0.25q=7.30\) β†’ \(0.15q=3.30\) β†’ \(q=22\)
22 quarters, 18 dimes

Problem #41 β€” Commission

Advanced

Earn $55/visit + 25% commission on $289 "gold seal" plans. 3 visits, earned $309.50. How many gold seal plans sold?

\(3(55)+0.25(289)g=309.50\) β†’ \(165+72.25g=309.50\) β†’ \(g=2\). 2 plans

Problem #45 β€” Acid Mixture

Advanced

Mix 8 liters of 40% acid with 70% acid to get 50% acid. How many liters of 70%?

Let \(x\) = liters of 70% solution.
\(0.40(8) + 0.70x = 0.50(8+x)\)
\(3.2 + 0.70x = 4 + 0.50x\)
\(0.20x = 0.8\) β†’ \(\mathbf{x = 4 \text{ liters}}\)

Problem #49 β€” Beverages System

Advanced

2 cappuccinos + 3 lattes = $14.55; 1 cappuccino + 2 lattes = $8.77. Price of each?

Multiply eq.2 by 2: \(2c+4l=17.54\). Subtract eq.1: \(l=2.99\). Then \(c=8.77-2(2.99)=2.79\).
Cappuccino $2.79, Latte $2.99

Problem #50 β€” Amusement Park

Advanced

2 adults + 3 children = $170. 3 adults + 5 children = $270. Ticket prices?

Multiply eq.1 by 3, eq.2 by 2: \(6a+9c=510\) and \(6a+10c=540\). Subtract: \(c=30\). Then \(2a=80\) β†’ \(a=40\).
Adult $40, Child $30
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Answered

Q1

Intro
Which set does \(-\sqrt{2}\) belong to?
\(\sqrt{2}\) cannot be expressed as \(p/q\) β€” it's irrational. Its negative is also irrational.

Q2

Intro
Evaluate \(8 \div 2 \cdot 4 - 7 + 3\).
Left to right: \(8\div2=4\), then \(4\times4=16\), then \(16-7+3=12\).

Q3

Intro
Simplify: \(2^{-1}\)
Negative exponent: \(a^{-n} = \frac{1}{a^n}\). So \(2^{-1} = \frac{1}{2}\).

Q4

Mid
Expand: \((x+7)(x-7)\)
Difference of squares: \((a+b)(a-b)=a^2-b^2\).

Q5

Mid
Solve: \(3x^2 + 5x = 0\)
Factor out \(x\): \(x(3x+5)=0\). Set each factor = 0.

Q6

Mid
Solve the inequality: \(-4y \leq 16\)
Dividing by a negative flips the inequality sign. \(-4y\leq16\) β†’ divide by \(-4\) β†’ flip β†’ \(y\geq-4\).

Q7

Mid
Find the slope through \((-2,2)\) and \((1,3)\).
\(m = \dfrac{3-2}{1-(-2)} = \dfrac{1}{3}\)

Q8

Advanced
Solve the system: \(y = 3x - 4\) and \(y = -2x + 11\)
Set equal: \(3x-4=-2x+11\) β†’ \(5x=15\) β†’ \(x=3\). Then \(y=3(3)-4=5\).