Solve |4x+12| > 16: Absolute Value Inequality Challenge

Q: Why do we get two separate inequalities from one absolute value inequality?

The absolute value $ |4x+12| $ represents distance from zero. For this distance to be greater than 16, the expression inside can be either very positive (> 16) or very negative (

Q: How do I know when to use 'or' versus 'and' in my final answer?

For absolute value inequalities like $ |A| > B $, always use OR because we want values that satisfy either condition. Use 'and' only when both conditions must be true simultaneously.

Q: What does the solution x > 1 or x < -7 look like on a number line?

Draw two separate regions: everything to the right of 1 (not including 1) and everything to the left of -7 (not including -7). The middle section from -7 to 1 is not part of the solution.

Q: Why isn't x = 1 or x = -7 included in the solution?

Because our inequality uses > (greater than), not ≥. At x = 1: $ |4(1)+12| = 16 $, which equals 16 but doesn't satisfy > 16.

Q: How can I check if my solution regions are correct?

Pick a test point from each region (like x = 2 and x = -8)Substitute into the original inequalityBoth should make $ |4x+12| > 16 $ truePick a point between the regions (like x = 0) - it should make the inequality false

Absolute Value Inequalities with Union Solutions

Given:

$\left|4x+12\right|>16$

Which of the following statements is necessarily true?

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Understand the problem

Given:

$\left|4x+12\right|>16$

Which of the following statements is necessarily true?

Step-by-step solution

To solve this inequality, we use the principle that if $|A| > B$ , then $A > B$ or $A < -B$ . Here's a detailed step-by-step solution:

Set up the two inequalities from the absolute value expression:
1. $4x + 12 > 16$
2. $4x + 12 < -16$
For the first inequality, $4x + 12 > 16$ :
- Subtract 12 from both sides to isolate the term with $x$ :
$4x > 16 - 12$
$4x > 4$
- Divide both sides by 4:
$x > 1$
For the second inequality, $4x + 12 < -16$ :
- Subtract 12 from both sides:
$4x < -16 - 12$
$4x < -28$
- Divide both sides by 4:
$x < -7$
The solution is the union of both results:
$x > 1$ or $x < -7$

Therefore, the correct answer among the given choices is $x > 1$ or $x < -7$ .

Final Answer

$x>1$ or $x<-7$

Key Points to Remember

Essential concepts to master this topic

Rule: If |A| > B, then A > B or A < -B
Technique: Solve 4x + 12 > 16 and 4x + 12 < -16 separately
Check: Test x = 2: |4(2) + 12| = |20| = 20 > 16 ✓

Common Mistakes

Avoid these frequent errors

Using 'and' instead of 'or' to connect solutions
Don't write x > 1 AND x < -7 = impossible condition! No number can be both greater than 1 and less than -7 simultaneously. Always use OR to connect the two solution regions for absolute value inequalities.

Practice Quiz

Test your knowledge with interactive questions

Find the absolute value inequality representation for:

$|x + 3| \leq 5$

FAQ

Everything you need to know about this question

Why do we get two separate inequalities from one absolute value inequality?

The absolute value $|4x+12|$ represents distance from zero. For this distance to be greater than 16, the expression inside can be either very positive (> 16) or very negative (< -16).

How do I know when to use 'or' versus 'and' in my final answer?

For absolute value inequalities like $|A| > B$ , always use OR because we want values that satisfy either condition. Use 'and' only when both conditions must be true simultaneously.

What does the solution x > 1 or x < -7 look like on a number line?

Draw two separate regions: everything to the right of 1 (not including 1) and everything to the left of -7 (not including -7). The middle section from -7 to 1 is not part of the solution.

Why isn't x = 1 or x = -7 included in the solution?

Because our inequality uses > (greater than), not ≥. At x = 1: $|4(1)+12| = 16$ , which equals 16 but doesn't satisfy > 16.

How can I check if my solution regions are correct?

Pick a test point from each region (like x = 2 and x = -8)
Substitute into the original inequality
Both should make $|4x+12| > 16$ true
Pick a point between the regions (like x = 0) - it should make the inequality false

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