Solve |3x - 4| ≤ 5: Absolute Value Inequality Solution

Q: Why does ≤ create an AND condition instead of OR?

Think of absolute value as distance from zero. When $ |3x - 4| \leq 5 $, you're saying the distance is at most 5, which means the expression must be between -5 and 5. That's an AND condition!

Q: How is this different from |3x - 4| ≥ 5?

Great question! $ |3x - 4| \geq 5 $ would give you OR conditions: either $ 3x - 4 \leq -5 $ OR $ 3x - 4 \geq 5 $. The ≥ creates two separate regions!

Q: Can I solve the two inequalities separately?

Yes, but remember they must both be true simultaneously. Solve $ 3x - 4 \leq 5 $ to get $ x \leq 3 $, and solve $ 3x - 4 \geq -5 $ to get $ x \geq -\frac{1}{3} $. Then find where both are true!

Q: How do I check if my interval is correct?

Pick test values! Try $ x = 0 $ (inside your interval): $ |3(0) - 4| = 4 \leq 5 ✓ $. Try $ x = 4 $ (outside): $ |3(4) - 4| = 8 > 5 ✗ $.

Q: What does the solution interval -1/3 ≤ x ≤ 3 mean visually?

On a number line, it's all values between and including $ -\frac{1}{3} $ and 3. Use closed circles at both endpoints since we have ≤ signs!

Absolute Value Inequalities with Compound Solutions

Given:

$\left|3x - 4\right| \leq 5$

Which of the following statements is necessarily true?

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given:

$\left|3x - 4\right| \leq 5$

Which of the following statements is necessarily true?

Step-by-step solution

To solve $\left| 3x - 4 \right| \leq 5$ , we should consider two scenarios for the absolute value: $3x - 4 \leq 5$ and $3x - 4 \geq -5$ .

1. Solving $3x - 4 \leq 5$ :

$3x - 4 \leq 5$

Add 4 to both sides:

$3x \leq 9$

Divide both sides by 3:

$x \leq 3$

2. Solving $3x - 4 \geq -5$ :

$3x - 4 \geq -5$

Add 4 to both sides:

$3x \geq -1$

Divide both sides by 3:

$x \geq -\frac{1}{3}$

Combining both results, we have $-\frac{1}{3} \leq x \leq 3$ , which is the correct answer.

Final Answer

$-\frac{1}{3} \leq x \leq 3$

Key Points to Remember

Essential concepts to master this topic

Rule: |expression| ≤ number creates two inequalities with AND
Technique: Rewrite |3x - 4| ≤ 5 as -5 ≤ 3x - 4 ≤ 5
Check: Test boundary values: x = -1/3 gives |3(-1/3) - 4| = |-5| = 5 ≤ 5 ✓

Common Mistakes

Avoid these frequent errors

Writing two separate OR conditions instead of AND
Don't solve |3x - 4| ≤ 5 as 3x - 4 ≤ 5 OR 3x - 4 ≥ -5 = gives wrong union! This misunderstands absolute value inequalities. Always use AND to create the intersection: -5 ≤ 3x - 4 ≤ 5 for one solution 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 does ≤ create an AND condition instead of OR?

Think of absolute value as distance from zero. When $|3x - 4| \leq 5$ , you're saying the distance is at most 5, which means the expression must be between -5 and 5. That's an AND condition!

How is this different from |3x - 4| ≥ 5?

Great question! $|3x - 4| \geq 5$ would give you OR conditions: either $3x - 4 \leq -5$ OR $3x - 4 \geq 5$ . The ≥ creates two separate regions!

Can I solve the two inequalities separately?

Yes, but remember they must both be true simultaneously. Solve $3x - 4 \leq 5$ to get $x \leq 3$ , and solve $3x - 4 \geq -5$ to get $x \geq -\frac{1}{3}$ . Then find where both are true!

How do I check if my interval is correct?

Pick test values! Try $x = 0$ (inside your interval): $|3(0) - 4| = 4 \leq 5 ✓$ . Try $x = 4$ (outside): $|3(4) - 4| = 8 > 5 ✗$ .

What does the solution interval -1/3 ≤ x ≤ 3 mean visually?

On a number line, it's all values between and including $-\frac{1}{3}$ and 3. Use closed circles at both endpoints since we have ≤ signs!

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Solve |3x - 4| ≤ 5: Absolute Value Inequality Solution

Absolute Value Inequalities with Compound Solutions

❤️ Continue Your Math Journey!

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why does ≤ create an AND condition instead of OR?

How is this different from |3x - 4| ≥ 5?

Can I solve the two inequalities separately?

How do I check if my interval is correct?

What does the solution interval -1/3 ≤ x ≤ 3 mean visually?

🌟 Unlock Your Math Potential

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