How to Solve: Simple Inequalities (GED)

How to Solve: Simple Inequalities (GED)

Score Impact: Simple inequalities appear 4-6 times on the GED Math test—mastering them can boost your score by 10-15 points, moving you from "Below Passing" to "Passing" or even "College Ready."

WHAT THIS QUESTION TYPE IS ACTUALLY TESTING

The GED isn’t testing whether you can solve x + 3 > 5. It’s testing: - Can you translate words into math? (e.g., "at least" → ≥, "no more than" → ≤) - Do you reverse the inequality sign when multiplying/dividing by a negative? (The #1 trap) - Can you eliminate wrong answers quickly under time pressure? (Most students waste time solving when they could guess strategically.)

ANATOMY OF THE QUESTION

Structure Breakdown

1. Stem: A word problem or direct inequality (e.g., "Solve for x: -2x + 5 ≤ 11")
2. Conditions: Hidden rules (e.g., "x must be a whole number")
3. Answer Choices: Usually 4 options, often including:
4. The correct solution
5. A "flipped sign" trap (if you forgot to reverse the inequality)
6. A "distributive error" trap (if you messed up expanding parentheses)
7. A "boundary error" (e.g., using < instead of ≤)

Representative Example Question

"A gym charges a $20 sign-up fee plus $15 per month. If a member pays no more than $110 total, which inequality represents the number of months (m) they can stay?" A) 20 + 15m ≤ 110 B) 20 + 15m < 110 C) 20 + 15m ≥ 110 D) 15m ≤ 110

(Correct answer: A. "No more than" = ≤, and the $20 fee is included.)

THE DECISION FRAMEWORK (Step-by-Step)

Run this every time—no exceptions.

1. Translate words → math.
2. Circle key phrases:
   • "At least" / "No less than" → ≥
   • "No more than" / "At most" → ≤
   • "More than" → >
   • "Less than" → <

3. Underline the variable and what it represents.

4. Isolate the variable.

5. Do the same steps as solving an equation (add/subtract first, then multiply/divide).

6. CRITICAL: If you multiply/divide by a negative number, flip the inequality sign.

7. Check boundary conditions.

8. If the inequality is ≤ or ≥, the boundary number (e.g., x = 3 in x ≤ 3) is included.

9. If it’s < or >, the boundary is not included.

10. Eliminate wrong answers.

11. Cross out options that:

    • Have the wrong inequality sign (e.g., < instead of ≤).
    • Ignore a term (e.g., forgetting the $20 fee in the gym example).
    • Flip the sign incorrectly (e.g., -2x > 6 → x < -3, not x > -3).

12. Test a number if unsure.

13. Pick a number that fits the inequality and plug it into the answer choices.
14. Example: For x + 3 > 5, test x = 3 (should work) and x = 2 (shouldn’t).

Worked Examples

Example 1 – Straightforward

Question: Solve for x: 4x - 7 < 13 Answer Choices: A) x < 5 B) x > 5 C) x ≤ 5 D) x ≥ 5

Step-by-Step: 1. Translate: Already in math form. No words to convert. 2. Isolate x:
- 4x - 7 < 13
- +7 to both sides: 4x < 20
- ÷4: x < 5 (No sign flip—dividing by positive 4.) 3. Check boundary: x = 5 is not included (because it’s <, not ≤). 4. Eliminate:
- B, D: Wrong direction (x > or ≥ 5).
- C: Includes x = 5 (but boundary isn’t included). 5. Answer: A

Example 2 – Common Trap (Negative Coefficient)

Question: Solve for y: -3y + 2 ≥ 14 Answer Choices: A) y ≤ -4 B) y ≥ -4 C) y ≤ 4 D) y ≥ 4

Step-by-Step: 1. Translate: Already in math form. 2. Isolate y:
- -3y + 2 ≥ 14
- -2: -3y ≥ 12
- ÷-3: y ≤ -4 (Sign flips because dividing by negative!) 3. Check boundary: y = -4 is included (because it’s ≤). 4. Eliminate:
- B, D: Wrong direction (y ≥).
- C: Wrong boundary (y ≤ 4 instead of -4). 5. Answer: A

Trap: Most students forget to flip the sign, picking B instead of A.

Example 3 – Hard Variant (Word Problem + Negative)

Question: "A number decreased by 8 is at least -5. Which inequality represents this?" Answer Choices: A) x - 8 ≥ -5 B) x - 8 ≤ -5 C) 8 - x ≥ -5 D) x + 8 ≥ -5

Step-by-Step: 1. Translate:
- "A number" = x
- "Decreased by 8" = x - 8
- "Is at least" = ≥
- So: x - 8 ≥ -5 2. Isolate x:
- x - 8 ≥ -5
- +8: x ≥ 3 (No sign flip—adding.) 3. Check boundary: x = 3 is included. 4. Eliminate:
- B: Wrong sign (≤ instead of ≥).
- C: Backwards (8 - x instead of x - 8).
- D: Wrong operation (+8 instead of -8). 5. Answer: A

Trap: Students misread "decreased by" as "8 - x" (C) or add instead of subtract (D).

WRONG ANSWER PATTERNS

Common Mistakes

TIME STRATEGY

• Target time: 45–60 seconds per question.
• When to skip: If you’re stuck after 90 seconds, flag it and move on.
• Minimum work to answer confidently:
• Translate words → math (10 sec).
• Solve for the variable (20 sec).
• Eliminate 2 wrong answers (15 sec).
• Test one number if unsure (10 sec).

BACKSOLVING AND SHORTCUTS

1. Plug in answer choices.
2. For x + 3 > 5, test x = 2 (A) and x = 3 (B). Only x = 3 works, so B is correct.
3. Use extreme numbers.
4. For 2x - 1 < 7, test x = 100 (way too big) and x = 0 (fits). Narrow down.
5. Eliminate first.
6. If two answers have the same boundary (e.g., x < 5 and x ≤ 5), the question likely tests boundary inclusion.
7. Look for sign flips.
8. If the answer choices include both x > -2 and x < -2, the question probably involves a negative coefficient.

1-Minute Recap

"Here’s the exact process to solve any inequality on the GED—fast and error-free:

1. Translate first. Circle words like ‘at least’ or ‘no more than’ and write the correct symbol (≥ or ≤).
2. Solve like an equation, but flip the sign if you multiply or divide by a negative number. No exceptions.
3. Check the boundary. If it’s ≤ or ≥, the number is included. If it’s < or >, it’s not.
4. Eliminate wrong answers by testing one number. Pick a value that fits the inequality and see which choices work.
5. If stuck, guess strategically. The GED loves trapping you with sign flips and boundary errors—so if two answers look similar, one of them is probably the trap.

This isn’t about doing more math—it’s about making fewer mistakes. Run the framework every time, and you’ll get these right in under a minute. Now go practice!

Final Tip: On test day, write this on your scratch paper: - ≤ = included - < = not included - Negative? Flip the sign!