Solve |5x + 3| ≥ 2x + 7: Absolute Value Inequality Analysis

Q: Why do I need two separate cases for absolute value inequalities?

The absolute value $ |5x + 3| $ can be either positive or negative inside. Case 1 handles when $ 5x + 3 \geq 0 $, and Case 2 handles when \( 5x + 3 .

Q: How do I know when to use ≥ or ≤ in my cases?

For $ |A| \geq B $, Case 1 is $ A \geq B $ and Case 2 is $ A \leq -B $. The inequality direction stays the same in Case 1 but flips the expression in Case 2.

Q: Why is the final answer x ≥ 4/3 AND x ≤ -10/7?

These represent two separate solution regions! The word 'AND' here means both conditions must be satisfied, giving us solutions in either $ x \geq \frac{4}{3} $ or $ x \leq -\frac{10}{7} $.

Q: How can I verify my solution is correct?

Pick test values from each solution region. Try $ x = 2 $ (from $ x \geq \frac{4}{3} $) and $ x = -2 $ (from $ x \leq -\frac{10}{7} $) in the original inequality.

Q: What if I get confused about the negative sign distribution?

Remember: $ -(2x + 7) = -2x - 7 $, not $ -2x + 7 $! Write out each step carefully and double-check that you've distributed the negative to every single term.

Absolute Value Inequalities with Two-Case Analysis

Given:

$\left|5x + 3\right| \geq 2x + 7$

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:

$\left|5x + 3\right| \geq 2x + 7$

Which of the following statements is necessarily true?

Step-by-step solution

Consider the inequality: $\left|5x + 3\right| \geq 2x + 7$ .

This inequality divides into two cases:

(1) $5x + 3 \geq 2x + 7$ and (2) $5x + 3 \leq -(2x + 7)$ .

For inequality (1):

$5x + 3 \geq 2x + 7$

Subtract $2x$ from both sides:

$3x + 3 \geq 7$

Subtract 3 from both sides:

$3x \geq 4$

Divide both sides by 3:

$x \geq \frac{4}{3}$

For inequality (2):

$5x + 3 \leq -2x - 7$

Add $2x$ to both sides:

$7x + 3 \leq -7$

Subtract 3 from both sides:

$7x \leq -10$

Divide both sides by 7:

$x \leq -\frac{10}{7}$

The solution sets are $x \geq \frac{4}{3}$ and $x \leq -\frac{10}{7}$ .

Final Answer

$x \geq \frac{4}{3}$ and $x \leq -\frac{10}{7}$

Key Points to Remember

Essential concepts to master this topic

Rule: Split absolute value inequality into two separate cases
Technique: Case 1: $5x + 3 \geq 2x + 7$ , Case 2: $5x + 3 \leq -(2x + 7)$
Check: Test boundary values: $x = \frac{4}{3}$ gives $|8| \geq \frac{22}{3}$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to distribute the negative sign in Case 2
Don't write $5x + 3 \leq -2x + 7$ instead of $5x + 3 \leq -2x - 7$ = wrong inequality! The negative must distribute to both terms inside the parentheses. Always distribute the negative sign to every term: $-(2x + 7) = -2x - 7$ .

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 two separate cases for absolute value inequalities?

The absolute value $|5x + 3|$ can be either positive or negative inside. Case 1 handles when $5x + 3 \geq 0$ , and Case 2 handles when $5x + 3 < 0$ .

How do I know when to use ≥ or ≤ in my cases?

For $|A| \geq B$ , Case 1 is $A \geq B$ and Case 2 is $A \leq -B$ . The inequality direction stays the same in Case 1 but flips the expression in Case 2.

Why is the final answer x ≥ 4/3 AND x ≤ -10/7?

These represent two separate solution regions! The word 'AND' here means both conditions must be satisfied, giving us solutions in either $x \geq \frac{4}{3}$ or $x \leq -\frac{10}{7}$ .

How can I verify my solution is correct?

Pick test values from each solution region. Try $x = 2$ (from $x \geq \frac{4}{3}$ ) and $x = -2$ (from $x \leq -\frac{10}{7}$ ) in the original inequality.

What if I get confused about the negative sign distribution?

Remember: $-(2x + 7) = -2x - 7$ , not $-2x + 7$ ! Write out each step carefully and double-check that you've distributed the negative to every single term.

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Solve |5x + 3| ≥ 2x + 7: Absolute Value Inequality Analysis

Absolute Value Inequalities with Two-Case Analysis

❤️ 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 do I need two separate cases for absolute value inequalities?

How do I know when to use ≥ or ≤ in my cases?

Why is the final answer x ≥ 4/3 AND x ≤ -10/7?

How can I verify my solution is correct?

What if I get confused about the negative sign distribution?

🌟 Unlock Your Math Potential

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