How to Solve: Absolute Value (GED)

How to Solve: Absolute Value (GED)

Score Impact: Absolute value questions appear 4-6 times per GED Math test—mastering them can boost your score by 10-15 points, pushing you into the 165+ (College Ready) range.

WHAT THIS QUESTION TYPE IS ACTUALLY TESTING

The GED isn’t testing whether you can compute |x| = 5. It’s testing: ✅ Logical reasoning – Can you interpret absolute value as "distance from zero" and apply it to inequalities? ✅ Attention to structure – Can you spot when a question is asking for two solutions (e.g., |x| = 3 → x = 3 or x = -3)? ✅ Trap avoidance – Can you resist the urge to drop the absolute value bars without considering both positive and negative cases?

ANATOMY OF THE QUESTION

Structure Breakdown

1. Stem – The problem statement, often involving an equation or inequality with absolute value.
2. Example: "Solve for x: |2x – 5| = 7"
3. Conditions – Sometimes hidden (e.g., "x is a positive integer").
4. Answer Choices – Usually 4 options, with 2 correct solutions (if the question asks for all possible values).
5. What to Ignore – Distractors like:
6. Only solving for the positive case (e.g., 2x – 5 = 7 but forgetting 2x – 5 = -7).
7. Misapplying inequalities (e.g., |x| < 3 → x < 3 and x > -3, not x < 3 or x > -3).

Representative Example Question

"If |3x + 1| = 8, what are the possible values of x?" A) -3 only B) 7/3 only C) -3 and 7/3 D) -7/3 and 3

THE DECISION FRAMEWORK (Step-by-Step)

Run this process for every absolute value question.

Step 1: Identify the Absolute Value Expression

• Circle or underline the part inside the absolute value bars (e.g., |3x + 1|).
• Ask: "Is this an equation or inequality?"
• Equation (|A| = B): Two solutions (A = B and A = -B).
• Inequality (|A| < B or |A| > B): Rewrite as a compound inequality.

Step 2: Check for Validity

• If the right side is negative (e.g., |x| = -4), no solution exists (absolute value is always ≥ 0).
• If the right side is zero (e.g., |x| = 0), only one solution (x = 0).

Step 3: Split into Two Cases

• Case 1: Inside the absolute value = positive right side.
• Example: |3x + 1| = 8 → 3x + 1 = 8
• Case 2: Inside the absolute value = negative right side.
• Example: |3x + 1| = 8 → 3x + 1 = -8

Step 4: Solve Each Case Separately

• Solve Case 1 for x.
• Solve Case 2 for x.
• Write both solutions (if the question asks for "all possible values").

Step 5: Match to Answer Choices

• If the question asks for all solutions, eliminate choices that:
• Only include one solution.
• Include extra incorrect values.
• If the question asks for a specific solution (e.g., "which is not a solution?"), test each option.

Step 6: Verify (If Time Permits)

• Plug solutions back into the original equation to confirm.
• Example: For x = -3, |3(-3) + 1| = |-9 + 1| = |-8| = 8 ✔️

Worked Examples

Example 1 – Straightforward (Equation)

Question: Solve |2x – 4| = 10. Answer Choices: A) x = 7 only B) x = -3 only C) x = 7 and x = -3 D) x = 3 and x = -7

Step-by-Step Solution: 1. Identify: |2x – 4| = 10 → Equation, two cases. 2. Validity: Right side (10) is positive → proceed. 3. Case 1: 2x – 4 = 10
- 2x = 14 → x = 7 4. Case 2: 2x – 4 = -10
- 2x = -6 → x = -3 5. Solutions: x = 7 and x = -3 6. Match: Choice C.

Elimination Logic: - A and B only have one solution → eliminate. - D has x = 3 and x = -7 → plugging in: |2(3) – 4| = 2 ≠ 10 → eliminate.

Example 2 – Common Trap (Inequality)

Question: Solve |x + 5| < 3. Answer Choices: A) x < -2 or x > -8 B) -8 < x < -2 C) x < -8 or x > -2 D) -2 < x < 8

Step-by-Step Solution: 1. Identify: |x + 5| < 3 → Inequality, rewrite as compound. 2. Rewrite: -3 < x + 5 < 3 3. Solve:
- Subtract 5 from all parts: -8 < x < -2 4. Match: Choice B.

Trap Avoidance: - Mistake: Students often forget to flip the inequality when splitting. - Incorrect: x + 5 < 3 or x + 5 > -3 → leads to Choice A (wrong). - Correct: Must be both conditions (AND), not OR.

Example 3 – Hard Variant (Word Problem)

Question: A machine fills bottles with 16 oz of juice. The actual amount varies by up to 0.5 oz. Which inequality represents all possible amounts (A) of juice in a bottle? Answer Choices: A) |A – 16| ≤ 0.5 B) |A – 0.5| ≤ 16 C) |A – 16| ≥ 0.5 D) |A + 16| ≤ 0.5

Step-by-Step Solution: 1. Translate: "Varies by up to 0.5 oz" → distance from 16 is ≤ 0.5. 2. Absolute Value Form: |A – 16| ≤ 0.5 3. Match: Choice A.

Elimination Logic: - B: |A – 0.5| ≤ 16 → Allows A = 100 (too large) → eliminate. - C: |A – 16| ≥ 0.5 → Excludes 15.5 to 16.5 → eliminate. - D: |A + 16| ≤ 0.5 → A ≈ -16 (negative juice? No) → eliminate.

WRONG ANSWER PATTERNS

1. Only One Solution Given

• Why it looks right: Students solve only the positive case (e.g., |x| = 5 → x = 5).
• Why it’s wrong: Absolute value equations always have two solutions (unless the right side is zero).

2. Incorrect Inequality Direction

• Why it looks right: Students write |x| < 3 as x < 3 or x > -3.
• Why it’s wrong: |x| < 3 means x is between -3 and 3 (AND, not OR).

3. Sign Errors in Splitting

• Why it looks right: Students forget to negate the right side (e.g., |2x + 1| = 4 → 2x + 1 = 4 and 2x + 1 = 4).
• Why it’s wrong: The second case must be 2x + 1 = -4.

4. Misapplying to Word Problems

• Why it looks right: Students confuse |x – center| ≤ range with |x – range| ≤ center.
• Why it’s wrong: The center (e.g., 16 oz) is the reference point, not the range (0.5 oz).

Common Mistakes

1. Forgetting to Split into Two Cases

• Why it happens: Students treat |x| = 5 like a regular equation (x = 5).
• Correct approach: Always write two equations (x = 5 and x = -5).

2. Dropping Absolute Value Bars Too Early

• Why it happens: Students see |x| and immediately remove bars without considering the negative case.
• Correct approach: Never drop bars until you’ve set up both cases.

3. Mixing Up AND/OR in Inequalities

• Why it happens: Students confuse |x| < 3 (AND) with |x| > 3 (OR).
• Correct approach:
• |x| < a → -a < x < a (AND)
• |x| > a → x < -a or x > a (OR)

4. Ignoring Negative Right Sides

• Why it happens: Students don’t check if the right side is negative (e.g., |x| = -2 → no solution).
• Correct approach: If right side is negative, no solution exists.

5. Overcomplicating Word Problems

• Why it happens: Students panic and don’t translate "varies by up to 0.5 oz" into |A – 16| ≤ 0.5.
• Correct approach: Identify the center (16 oz) and range (0.5 oz), then write the inequality.

TIME STRATEGY

• Target Time: 45-60 seconds per question.
• When to Skip:
• If you’re stuck after 30 seconds, flag and move on.
• If the question involves complex fractions (e.g., |(2x + 1)/3| = 5), skip and return later.
• Minimum Work Needed:
• Split into two cases (or rewrite inequality).
• Solve both cases (or the compound inequality).
• Match to answer choices (eliminate obvious wrongs).

BACKSOLVING AND SHORTCUTS

1. Plug in Answer Choices (Backsolving)

• If the question asks for all solutions, test each choice in the original equation.
• Example: For |2x – 4| = 10, test x = 7 and x = -3.
• |2(7) – 4| = 10 ✔️
• |2(-3) – 4| = 10 ✔️ → Choice C is correct.

2. Number Substitution

• For inequalities, pick a number inside and outside the range to test.
• Example: For |x + 5| < 3, test x = -4 (inside) and x = -10 (outside).
• |-4 + 5| = 1 < 3 ✔️
• |-10 + 5| = 5 < 3 ❌ → Confirms -8 < x < -2.

3. Elimination First

• Eliminate choices that:
• Only have one solution (for equations).
• Have the wrong inequality direction (e.g., |x| < 3 → x < 3 or x > -3).
• Include impossible values (e.g., negative juice amounts).

1-Minute Recap

"Here’s the 30-second rule for absolute value questions on the GED:

1. Is it an equation or inequality?
2. Equation (|A| = B) → two cases: A = B and A = -B.
3. Inequality (|A| < B) → rewrite: -B < A < B.

4. Inequality (|A| > B) → split: A < -B or A > B.

5. Check the right side:

6. If negative → no solution.

7. If zero → one solution (A = 0).

8. Solve both cases (or the compound inequality).

9. Match to answer choices—eliminate the ones that:
10. Only give one solution.
11. Have the wrong inequality direction.
12. Include impossible values.

That’s it. No shortcuts, no guessing—just split, solve, and match. Now go crush those 4-6 questions on test day!

Final Tip:

Absolute value = distance from zero. If you remember that, you’ll never mix up the cases. Split, solve, and move on. ?