GED Mathematical reasoning 2 — Questions and Answers
Question 1: Solve for x: 3x − 7 = 14
- x = 3
- x = 5
- x = 7 (Correct answer)
- x = 9
Correct answer: x = 7
Add 7 to both sides: 3x = 21. Divide both sides by 3: x = 7.
To solve a linear equation, isolate the variable by performing inverse operations.
Step 1: Add 7 to both sides to undo the subtraction.
3x − 7 + 7 = 14 + 7
3x = 21
Step 2: Divide both sides by 3 to undo the multiplication.
3x ÷ 3 = 21 ÷ 3
x = 7
Verification: substitute x = 7 back into the original equation: 3(7) − 7 = 21 − 7 = 14 ✓
Always verify your answer by substituting back into the original equation. This catches arithmetic errors and confirms your solution is correct.
Question 2: Which equation represents the line passing through (0, −3) with a slope of 2?
- y = −3x + 2
- y = 2x − 3 (Correct answer)
- y = 2x + 3
- y = −2x − 3
Correct answer: y = 2x − 3
Using slope-intercept form y = mx + b, slope m = 2 and y-intercept b = −3 gives y = 2x − 3.
The slope-intercept form of a linear equation is y = mx + b, where:
• m = slope (rate of change)
• b = y-intercept (where the line crosses the y-axis)
Given information:
• Slope (m) = 2
• The line passes through (0, −3), meaning the y-intercept (b) = −3
Substituting: y = 2x + (−3) = 2x − 3.
You can verify by checking the point (0, −3): y = 2(0) − 3 = −3 ✓
The slope of 2 means for every 1 unit you move right, you move 2 units up. So another point on the line would be (1, −1): y = 2(1) − 3 = −1 ✓
Question 3: Simplify the expression: 4(2x − 3) + 5x
- 13x − 12 (Correct answer)
- 13x − 3
- 8x − 12 + 5x
- 13x + 12
Correct answer: 13x − 12
Distribute: 4(2x − 3) = 8x − 12. Then combine like terms: 8x + 5x − 12 = 13x − 12.
Simplifying algebraic expressions requires two key steps: distribution and combining like terms.
Step 1 — Distribute 4 across (2x − 3):
4 × 2x = 8x
4 × (−3) = −12
So 4(2x − 3) = 8x − 12
Step 2 — Rewrite the full expression:
8x − 12 + 5x
Step 3 — Combine like terms (terms with the same variable):
8x + 5x = 13x (x terms together)
−12 stays (constant term)
Result: 13x − 12
Note: Answer choice C shows the work halfway done but is not fully simplified. The GED expects fully simplified answers.
Question 4: A function is defined as f(x) = x² − 4x + 3. What is f(2)?
Correct answer: −1
f(2) = (2)² − 4(2) + 3 = 4 − 8 + 3 = −1.
To evaluate a function at a specific value, substitute that value for x everywhere in the expression.
f(x) = x² − 4x + 3
f(2) = (2)² − 4(2) + 3
Following order of operations:
1. Exponent: (2)² = 4
2. Multiplication: 4(2) = 8
3. Substituting: 4 − 8 + 3
4. Left to right: 4 − 8 = −4, then −4 + 3 = −1
f(2) = −1
Note that x = 2 is between the two roots of this quadratic. You can factor f(x) = (x−1)(x−3), which has roots at x = 1 and x = 3. At x = 2 (the midpoint), the parabola reaches its minimum value of −1.
Question 5: Solve the inequality: 2x + 5 > 13
- x > 4 (Correct answer)
- x > 9
- x < 4
- x ≥ 4
Correct answer: x > 4
Subtract 5: 2x > 8. Divide by 2: x > 4. The direction of the inequality does not change when dividing by a positive number.
Solving inequalities follows the same steps as solving equations, with one critical rule: if you multiply or divide both sides by a NEGATIVE number, you must flip the inequality sign.
Step 1: Subtract 5 from both sides.
2x + 5 − 5 > 13 − 5
2x > 8
Step 2: Divide both sides by 2 (positive, so no flip).
x > 4
This means any value greater than 4 (but NOT including 4) is a solution. On a number line, draw an open circle at 4 and shade to the right.
Verification: Try x = 5: 2(5) + 5 = 15 > 13 ✓
Try x = 4: 2(4) + 5 = 13, which is NOT > 13 ✓ (4 is correctly excluded)
Question 6: If y varies directly with x, and y = 15 when x = 5, what is y when x = 8?
- 20
- 24 (Correct answer)
- 25
- 40
Correct answer: 24
Direct variation: y = kx. Find k: 15 = k(5), so k = 3. Then y = 3(8) = 24.
Direct variation means y = kx, where k is the constant of variation (also called the constant of proportionality).
Step 1: Find k using the given values (y = 15, x = 5).
y = kx → 15 = k(5) → k = 15/5 = 3
Step 2: Use k = 3 to find y when x = 8.
y = 3(8) = 24
Alternatively, use proportions since y/x is constant:
15/5 = y/8
3 = y/8
y = 24 ✓
Direct variation graphs are straight lines through the origin (0, 0) with slope k. In this case, the line y = 3x passes through (0,0), (5,15), and (8,24).
