🦕 Dino Rex's Pre-Algebra
Adventures in Math — Let's Roar Through Numbers!
✦ ✦
Hi! I'm Dino Rex! 🦕
Let's conquer Pre-Algebra together! Are you ready to ROAR? 🌟
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📚 Concepts & Formulas
🔢 1. Order of Operations
Evaluate expressions step by step using the correct order.
PEMDAS: Parentheses → Exponents → Multiply/Divide → Add/Subtract
Left to right for × ÷ and + −. Never skip steps!
✏️ Example
3 + 4 × 2 = 3 + 8 = 11
(Do × before +)
➕ 2. Integer Operations
Rules for adding, subtracting, multiplying, and dividing integers.
Same signs → Add & keep | Different signs → Subtract & use larger sign
(+)(+)=+ | (−)(−)=+ | (+)(−)=− | (−)(+)=−
Negative × Negative = POSITIVE! Two wrongs make a right in math!
✏️ Example
−8 + 5 = −3 (different signs: 8−5=3, larger is −8 so answer is −)
(−3) × (−4) = +12
🍕 3. Fractions
Add/Subtract: need same denominator. Multiply: straight across. Divide: flip and multiply.
a/b + c/d = (ad + bc) / bd
(a/b) ÷ (c/d) = (a/b) × (d/c)
KCF: Keep, Change, Flip for division!
✏️ Example
1/2 + 1/3 = 3/6 + 2/6 = 5/6
(3/4) ÷ (1/2) = (3/4) × (2/1) = 3/2 = 1½
📐 4. Variables & Equations
Solve for the unknown by isolating the variable using inverse operations.
Whatever you do to one side, do to the other side!
GOAL: Get the variable ALONE on one side.
✏️ Example
2x + 3 = 11
Subtract 3: 2x = 8
Divide by 2: x = 4
📊 5. Ratios, Proportions & Percent
A proportion is two equal ratios. Cross-multiply to solve. Percent means "per 100."
a/b = c/d → a×d = b×c
Percent = (part ÷ whole) × 100
Cross-multiply to crack proportions!
✏️ Example
3/4 = x/12 → 4x = 36 → x = 9
20% of 80 = 0.20 × 80 = 16
⚡ 6. Exponents & Square Roots
Exponent means repeated multiplication. Square root is the reverse.
aⁿ = a × a × a … (n times)
a⁻ⁿ = 1/aⁿ | √(a²) = a (for a ≥ 0)
Negative exponent = flip to denominator: 3⁻² = 1/9
✏️ Example
2³ = 2×2×2 = 8
√49 = 7 (because 7×7=49)
3⁻² = 1/3² = 1/9
📏 7. Algebraic Expressions
Combine like terms (same variable & same power). Use the Distributive Property to expand.
a(b + c) = ab + ac
Like terms: 3x + 5x = 8x (add coefficients)
ONLY combine like terms — you can't add 3x + 5y!
✏️ Example
5x + 3x − 2x = 6x
3(x + 4) = 3x + 12
📉 8. Inequalities, Absolute Value & More
Inequalities: flip the sign when multiplying/dividing by negative. |a| = distance from 0.
|a| = a if a ≥ 0 ; |a| = −a if a < 0
Scientific notation: a × 10ⁿ (1 ≤ a < 10)
Flip inequality sign when multiplying/dividing by a NEGATIVE!
✏️ Example
|−9| = 9
x + 3 > 7 → x > 4
0.0035 = 3.5 × 10⁻³
🎯 20 Challenge Problems
1
Order of Operations
What is the value of the expression?
3 + 4 × 2
2
Integer Operations
Evaluate:
−8 + 5
3
Fractions
Add the fractions:
1/2 + 1/3
4
Solving Equations
Solve for x:
2x + 3 = 11
5
Proportions
Solve the proportion:
3/4 = x/12
6
Percentages
What is 20% of 80?
20% × 80 = ?
7
Exponents
Evaluate the exponent:
2³ = ?
8
Square Roots
Find the square root:
√49 = ?
9
Distributive Property
Expand using the Distributive Property:
3(x + 4) = ?
10
Inequalities
Solve the inequality:
x + 3 > 7
11
Absolute Value
Find the absolute value:
|−9| = ?
12
Slope / Coordinate Plane
What is the slope of the line passing through (0, 0) and (2, 4)?
slope = rise/run = ?
13
Ratios
Simplify the ratio to its lowest terms:
12 : 8
14
Scientific Notation
Write in scientific notation:
0.0035
15
Mean (Average)
Find the mean of the data set:
{4, 8, 6, 2}
16
Perimeter
What is the perimeter of a rectangle with length 5 and width 3?
P = 2(l + w)
17
Combining Like Terms
Simplify the expression:
5x + 3x − 2x
18
Negative Exponents
Evaluate:
3⁻² = ?
19
Two-Step Equations
Solve for x:
2x − 5 = 9
20
Integer Multiplication
Evaluate:
(−3) × (−4) = ?
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Amazing!
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🔑 Answer Key & Full Solutions
Q1 — Order of Operations
Answer: D) 11
Step 1: Identify the operations: 3 + 4 × 2
Step 2: PEMDAS — Multiplication comes before Addition.
Step 3: Calculate 4 × 2 = 8 first.
Step 4: Then add: 3 + 8 = 11.
⚠️ Common error: doing left-to-right gives 7 × 2 = 14 — WRONG! Always multiply before adding.
Q2 — Integer Operations
Answer: B) −3
Step 1: Different signs (−8 is negative, +5 is positive).
Step 2: Subtract the smaller absolute value from the larger: 8 − 5 = 3.
Step 3: Keep the sign of the number with the larger absolute value: |−8| > |5|, so the answer is negative.
Result: −3
Q3 — Fractions
Answer: B) 5/6
Step 1: Find the Least Common Denominator (LCD) of 2 and 3: LCD = 6.
Step 2: Convert: 1/2 = 3/6 and 1/3 = 2/6.
Step 3: Add numerators: 3/6 + 2/6 = 5/6.
⚠️ Never add denominators: 1/2 + 1/3 ≠ 2/5.
Q4 — Solving Equations
Answer: C) x = 4
Step 1: Start: 2x + 3 = 11
Step 2: Subtract 3 from both sides: 2x = 11 − 3 = 8
Step 3: Divide both sides by 2: x = 8 ÷ 2 = 4
Check: 2(4) + 3 = 8 + 3 = 11 ✓
Q5 — Proportions
Answer: C) x = 9
Step 1: Cross-multiply: 3 × 12 = 4 × x
Step 2: 36 = 4x
Step 3: Divide both sides by 4: x = 36 ÷ 4 = 9
Check: 3/4 = 9/12 → both reduce to 3/4 ✓
Q6 — Percentages
Answer: A) 16
Step 1: Convert 20% to decimal: 20% = 20/100 = 0.20
Step 2: Multiply: 0.20 × 80 = 16
Alternative: 10% of 80 = 8, so 20% = 8 × 2 = 16 ✓
Q7 — Exponents
Answer: B) 8
Step 1: 2³ means 2 multiplied by itself 3 times.
Step 2: 2 × 2 = 4, then 4 × 2 = 8
⚠️ Common error: 2³ ≠ 2 × 3 = 6. Exponent means repeated multiplication, not multiplication by the exponent.
Q8 — Square Roots
Answer: C) 7
Step 1: Ask: "What number times itself equals 49?"
Step 2: 7 × 7 = 49, so √49 = 7
Memory tip: Perfect squares to memorize: 1,4,9,16,25,36,49,64,81,100.
Q9 — Distributive Property
Answer: A) 3x + 12
Step 1: Multiply 3 by each term inside the parentheses.
Step 2: 3 × x = 3x
Step 3: 3 × 4 = 12
Result: 3x + 12. ⚠️ Remember to distribute to BOTH terms inside!
Q10 — Inequalities
Answer: B) x > 4
Step 1: x + 3 > 7
Step 2: Subtract 3 from both sides: x > 7 − 3
Step 3: x > 4
⚠️ The inequality sign only flips when you multiply or divide by a negative number. Here we subtracted, so it stays the same.
Q11 — Absolute Value
Answer: A) 9
Step 1: Absolute value means distance from 0 on a number line — always non-negative.
Step 2: |−9| = 9, because −9 is 9 units away from 0.
Rule: |a| = a if a ≥ 0; |a| = −a if a < 0. So |−9| = −(−9) = 9.
Q12 — Slope
Answer: C) 2
Step 1: Formula: slope = (y₂ − y₁) / (x₂ − x₁)
Step 2: Points: (0, 0) and (2, 4)
Step 3: slope = (4 − 0) / (2 − 0) = 4/2 = 2
Q13 — Ratios
Answer: B) 3 : 2
Step 1: Find the GCF of 12 and 8: GCF = 4.
Step 2: Divide both parts by 4: 12 ÷ 4 = 3, 8 ÷ 4 = 2.
Result: 3 : 2
Check: 6:4 is simplified from 12:8 but not fully reduced (still divisible by 2).
Q14 — Scientific Notation
Answer: C) 3.5 × 10⁻³
Step 1: Move the decimal point until you have a number between 1 and 10: 0.0035 → 3.5
Step 2: Count how many places you moved right: 3 places to the right.
Step 3: Moving right means negative exponent: 3.5 × 10⁻³
⚠️ The coefficient must be between 1 and 10, so 35 × 10⁻⁴ and 0.35 × 10⁻² are NOT proper scientific notation.
Q15 — Mean
Answer: D) 5
Step 1: Mean = sum of all values ÷ number of values.
Step 2: Sum: 4 + 8 + 6 + 2 = 20
Step 3: Number of values: 4
Step 4: Mean = 20 ÷ 4 = 5
Q16 — Perimeter
Answer: B) 16
Step 1: Formula for perimeter of rectangle: P = 2(l + w)
Step 2: P = 2(5 + 3) = 2(8) = 16
⚠️ A is 15 = area (l × w), not perimeter. Don't confuse area and perimeter!
Q17 — Combining Like Terms
Answer: C) 6x
Step 1: All three terms are "like terms" because they all have the variable x to the first power.
Step 2: Add and subtract the coefficients: 5 + 3 − 2 = 6
Step 3: Keep the variable: 6x
Q18 — Negative Exponents
Answer: B) 1/9
Step 1: Rule: a⁻ⁿ = 1/aⁿ
Step 2: 3⁻² = 1/3² = 1/(3 × 3) = 1/9
⚠️ A negative exponent does NOT make the answer negative! It means "take the reciprocal."
Q19 — Two-Step Equations
Answer: C) x = 7
Step 1: 2x − 5 = 9
Step 2: Add 5 to both sides: 2x = 9 + 5 = 14
Step 3: Divide both sides by 2: x = 14 ÷ 2 = 7
Check: 2(7) − 5 = 14 − 5 = 9 ✓
Q20 — Integer Multiplication
Answer: B) 12
Step 1: Multiply the absolute values: 3 × 4 = 12
Step 2: Determine the sign: Negative × Negative = POSITIVE
Result: (−3) × (−4) = +12
Memory: Two negatives = positive! Think of it as "the enemy of my enemy is my friend."