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Core Concepts & Formulas
1 · Number Properties & Integers
Integers include all whole numbers and their negatives: …−3, −2, −1, 0, 1, 2, 3…absolute value |x| is the distance from zero — always ≥ 0.
|−7| = 7 |5| = 5 |0| = 0
Adding same signs → keep sign & add. Opposite signs → subtract & keep sign of larger absolute value.
Example: (−4) + 9 = +5 (because |9| > |−4|, keep positive, 9−4=5)
2 · Fractions, Decimals & Percents
Fraction ↔ Decimal: divide numerator by denominator.Decimal ↔ Percent: multiply by 100 (move point 2 right).
3/4 = 0.75 = 75% 2/5 = 0.4 = 40%
% means "per hundred." So 35% = 35/100 = 0.35
Example: What is 30% of 80? → 0.30 × 80 = 24
3 · Order of Operations (PEMDAS)
Always solve in this order to get the correct answer.
P · Parentheses → E · Exponents → M/D · Multiply & Divide (left→right) → A/S · Add & Subtract (left→right)
Rhyme: "Please Excuse My Dear Aunt Sally" 🧮
Example: 3 + 4 × 2² − 1 = 3 + 4×4 − 1 = 3 + 16 − 1 = 18
4 · Variables, Expressions & Equations
A variable is a letter holding an unknown value. An expression has no equals sign. An equation has an equals sign — solve by keeping both sides balanced.
Expression: 3x + 5 Equation: 3x + 5 = 17 → x = 4
Whatever you do to one side, do to the other side!
Example: 2x − 3 = 7 → 2x = 10 → x = 5
5 · Inequalities
Solve like equations BUT flip the inequality sign when multiplying or dividing by a negative number.
−2x < 6 → divide by −2, flip sign → x > −3
Multiply/divide by negative? FLIP the sign! < becomes >
Example: 3x − 1 > 8 → 3x > 9 → x > 3
6 · Ratios & Proportions
A ratio compares two quantities. A proportion says two ratios are equal. Use cross-multiplication to solve.
a/b = c/d → a × d = b × c
Cross-multiply to solve proportions: "butterfly wings"
Example: x/6 = 4/8 → 8x = 24 → x = 3
7 · Exponents & Square Roots
bn means multiply b by itself n times. √x asks: what number squared gives x?
2⁵ = 2×2×2×2×2 = 32 √49 = 7 (since 7² = 49)
Perfect squares: 1,4,9,16,25,36,49,64,81,100,121,144
Example: 3³ = 27 √64 = 8
8 · Linear Equations & Slope
Slope-intercept form: y = mx + b , where m = slope and b = y-intercept.
slope m = (y₂ − y₁) / (x₂ − x₁) = rise / run
y = mx + b: m = slope (steepness), b = where line crosses y-axis
Example: Line through (0, 3) and (2, 7): m = (7−3)/(2−0) = 4/2 = 2. Equation: y = 2x + 3
A 7B −23C 23D −7
✅ Correct! Great work!
Absolute value is always the positive distance from zero.
|−15| = 15
|−8| = 8
15 + 8 = 23 ✓
A −27B 21C 27D −21
✅ Correct! Great work!
Use order of operations — multiply first, then add.
(−6) × (−4) = +24 (negative × negative = positive)
24 + (−3) = 24 − 3 = 21 ✓
A 37.5%B 38%C 30%D 83%
✅ Correct! Great work!
Divide numerator by denominator, then multiply by 100.
3 ÷ 8 = 0.375
0.375 × 100 = 37.5% ✓
A $90B $95C $30D $100
✅ Correct! Great work!
First find the discount amount, then subtract from original.
Discount = 25% × $120 = 0.25 × 120 = $30
Sale price = $120 − $30 = $90 ✓
A 5B 3C 11D 9
✅ Correct! Great work!
Follow PEMDAS carefully step by step.
P: (8 − 5) = 3
E: 3² = 9
M/D: 3 × 9 ÷ 9 = 27 ÷ 9 = 3
A/S: 2 + 3 = 5 ✓
A 27B 11C −3D 3
✅ Correct! Great work!
Exponents first, then multiply/divide left-to-right, then subtract.
E: 4² = 16
M/D left→right: 18 ÷ 2 = 9, then 9 × 3 = 27
A/S: 27 − 16 = 11 ✓
A x = 23B x = 5C x = −5D x = −23
✅ Correct! Great work!
Subtract 14 from both sides to isolate x.
x + 14 = 9
x + 14 − 14 = 9 − 14
x = −5 ✓
Check: (−5) + 14 = 9 ✓
A x = 3.5B x = 7C x = 8D x = 14
✅ Correct! Great work!
Add 7 to both sides first, then divide by 4.
4x − 7 = 21
4x − 7 + 7 = 21 + 7 → 4x = 28
4x ÷ 4 = 28 ÷ 4 → x = 7 ✓
Check: 4(7) − 7 = 28 − 7 = 21 ✓
A x = 5B x = 3C x = 13D x = −1
✅ Correct! Great work!
Distribute, then collect like terms, then solve.
3(x + 4) = 2x + 17
3x + 12 = 2x + 17 (distribute)
3x − 2x = 17 − 12 (move x to left, constants to right)
x = 5 ✓
Check: 3(5+4) = 3(9) = 27; 2(5)+17 = 27 ✓
A x > −3B x < −3C x < 3D x > 3
✅ Correct! Great work!
Subtract 6, then divide by −3 (FLIP the sign!).
−3x + 6 > 15
−3x > 15 − 6 → −3x > 9
x < 9 ÷ (−3) [flip sign when dividing by negative]
x < −3 ✓
Check: x = −4 (should work): −3(−4)+6 = 12+6 = 18 > 15 ✓
A x = 4B x = 5C x = 3D x = −1
✅ Correct! Great work!
Solve for x in the compound inequality by adding 3 then dividing by 2.
−1 ≤ 2x − 3 < 5
−1 + 3 ≤ 2x < 5 + 3 → 2 ≤ 2x < 8
1 ≤ x < 4
So x must satisfy 1 ≤ x < 4. Check options:
x = 3: 1 ≤ 3 < 4 ✓ x = 4: 4 < 4 is false ✗ x = 5: 5 < 4 false ✗ x = −1: −1 < 1 false ✗
A $5.60B $4.80C $6.00D $7.00
✅ Correct! Great work!
Set up a proportion and cross-multiply.
5/3.50 = 8/x
5x = 3.50 × 8 = 28
x = 28 ÷ 5 = $5.60 ✓
A 55 mphB 70 mphC 60 mphD 65 mph
✅ Correct! Great work!
Unit rate = total miles ÷ total hours.
210 ÷ 3.5 = 60 mph ✓
(Or: 210 ÷ 3.5 = 2100 ÷ 35 = 60)
A 2¹²B 4⁷C 2⁷D 2⁸
✅ Correct! Great work!
When multiplying same base, add the exponents: b^m × b^n = b^(m+n)
2³ × 2⁴ = 2^(3+4) = 2⁷ ✓
2⁷ = 128 (you can verify: 8 × 16 = 128)
A 6B 9C 8D 4
✅ Correct! Great work!
Find the approximate values first.
√49 = 7, so √50 ≈ 7.07
√81 = 9, so √80 ≈ 8.94
We need a value between 7.07 and 8.94
8 is between 7.07 and 8.94 ✓ (8² = 64, and 50 < 64 < 80)
Check others: 6² = 36 < 50 ✗ 9² = 81 > 80 ✗ 4² = 16 < 50 ✗
A 3B 1/2C 2D 4
✅ Correct! Great work!
Use the slope formula: m = (y₂ − y₁) ÷ (x₂ − x₁)
m = (13 − 5) ÷ (6 − 2)
m = 8 ÷ 4 = 2 ✓
A y = 4x − 3B y = −3x + 4C y = 3x + 4D y = −3x − 4
✅ Correct! Great work!
Slope-intercept form is y = mx + b, where m = slope and b = y-intercept.
m = −3, b = 4
y = −3x + 4 ✓
A 14x − 15B 10x − 3C 14x − 3D 10x − 15
✅ Correct! Great work!
Distribute 5 into the parentheses, then combine like terms.
5(2x − 3) + 4x
= 10x − 15 + 4x (distribute)
= (10x + 4x) − 15 (combine like terms)
= 14x − 15 ✓
A 4/3B 0C 2D 1/3
✅ Correct! Great work!
Any nonzero number to the 0 power = 1. Negative exponent = reciprocal.
5⁰ = 1 (any base ≠ 0 raised to 0 = 1)
3⁻¹ = 1/3 (negative exponent means reciprocal)
1 + 1/3 = 3/3 + 1/3 = 4/3 ✓
A d = 3h + 15B d = 15h − 3C d = 15 − 3hD d = 3 − 15h
✅ Correct! Great work!
Rex starts at 15 miles and gets closer (decreasing distance) at 3 mph.
Start: d = 15 when h = 0
Each hour: distance decreases by 3 miles
d = 15 − 3h ✓
Check: after 1 hr: d = 15 − 3(1) = 12 miles (closer ✓); after 5 hrs: d = 0 (home!) ✓
💡 Answer Key & Full Explanations
Q1
C
23 — |−15| = 15, |−8| = 8; 15 + 8 = 23
Q2
B
21 — (−6)(−4) = +24; 24 + (−3) = 21
Q3
A
37.5% — 3÷8 = 0.375 = 37.5%
Q4
A
$90 — 25% of $120 = $30 discount; $120 − $30 = $90
Q5
A
5 — (8−5)=3; 3²=9; 3×9÷9=3; 2+3=5
Q6
B
11 — 4²=16; 18÷2=9; 9×3=27; 27−16=11
Q7
C
x = −5 — x + 14 = 9; x = 9 − 14 = −5
Q8
B
x = 7 — 4x = 21 + 7 = 28; x = 28÷4 = 7
Q9
A
x = 5 — 3x+12=2x+17; x=5
Q10
B
x < −3 — −3x > 9; divide by −3 and FLIP: x < −3
Q11
C
x = 3 — Solving gives 1 ≤ x < 4; only x=3 fits
Q12
A
$5.60 — 5/3.50 = 8/x; 5x=28; x=$5.60
Q13
C
60 mph — 210 ÷ 3.5 = 60
Q14
C
2⁷ — Same base: add exponents: 3+4=7
Q15
C
8 — √50≈7.07, √80≈8.94; 8 is between them (8²=64)
Q16
C
2 — m = (13−5)÷(6−2) = 8÷4 = 2
Q17
B
y = −3x + 4 — y=mx+b; m=−3, b=4
Q18
A
14x − 15 — 10x−15+4x = 14x−15
Q19
A
4/3 — 5⁰=1; 3⁻¹=1/3; 1+1/3=4/3
Q20
C
d = 15 − 3h — Starts at 15, decreases by 3 per hour
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