Solve |x-4| < 8: Absolute Value Inequality Analysis

Q: Why do I need to write two separate inequalities?

The absolute value \( |x - 4| means the distance from x to 4 is less than 8 units. This creates boundaries on both sides of 4, so you need two inequalities to capture both limits.

Q: How do I remember which way the inequality signs go?

Think of absolute value as distance. If \( |x - 4| , then x is within 8 units of 4. This means \( -8 , which splits into two parts naturally.

Q: What if I get confused about adding 4 to both sides?

Always do the same operation to all parts! For \( -8 , add 4 to all three parts: \( -8 + 4 gives \( -4 .

Q: How can I check if my answer interval is correct?

Pick any number from your interval and substitute it back! Try x = 0: $ |0 - 4| = 4 $, and since 4

Q: What's the difference between |x - 4| 8?

Less than ( Creates a single interval between two boundsGreater than (>): Creates two separate regions outside the boundsThe gives you two separate pieces.

Absolute Value Inequalities with Double Bounds

Given:

$\left|x-4\right|<8$

Which of the following statements is necessarily true?

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Understand the problem

Given:

$\left|x-4\right|<8$

Which of the following statements is necessarily true?

Step-by-step solution

To solve the inequality $|x - 4| < 8$ , we will break it down into two separate inequalities.

First, recognize that $|x - 4| < 8$ means the expression $x - 4$ can vary between -8 and 8 without violating the inequality constraint.
This gives us two inequalities to solve: $-8 < x - 4$ and $x - 4 < 8$ .

Let's solve each inequality:
1. For $-8 < x - 4$ :
- Add 4 to both sides to isolate $x$ :
$-8 + 4 < x \rightarrow -4 < x$ 2. For $x - 4 < 8$ :
- Add 4 to both sides to isolate $x$ :
$x < 8 + 4 \rightarrow x < 12$

By combining these results, we obtain the solution:
$-4 < x < 12$

Therefore, the range of $x$ that satisfies the inequality $|x - 4| < 8$ is $-4 < x < 12$ .

Hence, the correct statement from the given choices is $\boxed{-4 < x < 12}$ .

Final Answer

$-4 < x < 12$

Key Points to Remember

Essential concepts to master this topic

Rule: |x - 4| < 8 means -8 < x - 4 < 8
Technique: Split into two inequalities: -8 < x - 4 and x - 4 < 8
Check: Test x = 0: |0 - 4| = 4 < 8, and -4 < 0 < 12 ✓

Common Mistakes

Avoid these frequent errors

Only solving one side of the inequality
Don't solve just x - 4 < 8 and get x < 12 = incomplete solution! You miss the lower bound constraint and get answers outside the valid range. Always solve both -8 < x - 4 AND x - 4 < 8 to find the complete interval.

Practice Quiz

Test your knowledge with interactive questions

Given:

$\left|2x-1\right|>-10$

Which of the following statements is necessarily true?

FAQ

Everything you need to know about this question

Why do I need to write two separate inequalities?

The absolute value $|x - 4| < 8$ means the distance from x to 4 is less than 8 units. This creates boundaries on both sides of 4, so you need two inequalities to capture both limits.

How do I remember which way the inequality signs go?

Think of absolute value as distance. If $|x - 4| < 8$ , then x is within 8 units of 4. This means $-8 < x - 4 < 8$ , which splits into two parts naturally.

What if I get confused about adding 4 to both sides?

Always do the same operation to all parts! For $-8 < x - 4 < 8$ , add 4 to all three parts: $-8 + 4 < x < 8 + 4$ gives $-4 < x < 12$ .

How can I check if my answer interval is correct?

Pick any number from your interval and substitute it back! Try x = 0: $|0 - 4| = 4$ , and since 4 < 8, it works. Also check that -4 < 0 < 12 is true.

What's the difference between |x - 4| < 8 and |x - 4| > 8?

Less than (<): Creates a single interval between two bounds
Greater than (>): Creates two separate regions outside the bounds

The < version gives you one continuous range, while > gives you two separate pieces.

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Solve |x-4| < 8: Absolute Value Inequality Analysis

Absolute Value Inequalities with Double Bounds

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why do I need to write two separate inequalities?

How do I remember which way the inequality signs go?

What if I get confused about adding 4 to both sides?

How can I check if my answer interval is correct?

What's the difference between |x - 4| < 8 and |x - 4| > 8?

🌟 Unlock Your Math Potential

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