Solve |4x - 5| ≤ 2x + 9: Absolute Value Inequality Challenge

Q: Why do I need to split the absolute value into two cases?

Because the expression 4x - 5 inside the absolute value can be either positive or negative! When it's positive, |4x - 5| = 4x - 5. When it's negative, |4x - 5| = -(4x - 5). Both cases must be satisfied simultaneously.

Q: How do I know which case applies to which values of x?

You don't need to worry about that! Just solve both cases separately, then find where they overlap. The final answer is the intersection of both solutions.

Q: What does it mean to combine the solutions?

Take the intersection (overlap) of both cases. From Case 1: $ x \leq 7 $. From Case 2: $ x \geq -\frac{2}{3} $. Combined: $ -\frac{2}{3} \leq x \leq 7 $.

Q: Can I test my answer by plugging in values?

Absolutely! Try x = 0: $ |4(0) - 5| = 5 $ and $ 2(0) + 9 = 9 $. Since 5 ≤ 9, x = 0 works! Try values inside and outside your solution interval.

Q: What if the right side of the inequality is negative?

If the right side is negative, then the inequality has no solution! Absolute values are always non-negative, so |expression| can never be less than a negative number.

Absolute Value Inequalities with Split Cases

Given:

$\left|4x - 5\right| \leq 2x + 9$

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:

$\left|4x - 5\right| \leq 2x + 9$

Which of the following statements is necessarily true?

Step-by-step solution

Let's solve the inequality: $\left|4x - 5\right| \leq 2x + 9$ .

This splits into two cases:

(1) $4x - 5 \leq 2x + 9$ and (2) $4x - 5 \geq -(2x + 9)$ .

For inequality (1):

$4x - 5 \leq 2x + 9$

Subtract $2x$ from both sides:

$2x - 5 \leq 9$

Add 5 to both sides:

$2x \leq 14$

Divide both sides by 2:

$x \leq 7$

For inequality (2):

$4x - 5 \geq -2x - 9$

Add $2x$ to both sides:

$6x - 5 \geq -9$

Add 5 to both sides:

$6x \geq -4$

Divide both sides by 6:

$x \geq -\frac{2}{3}$

Combining both solutions gives us $-\frac{2}{3} \leq x \leq 7$ .

Final Answer

$-\frac{2}{3} \leq x \leq 7$

Key Points to Remember

Essential concepts to master this topic

Rule: Split |expression| ≤ value into two separate inequality cases
Technique: Case 1: 4x - 5 ≤ 2x + 9, Case 2: 4x - 5 ≥ -(2x + 9)
Check: Test x = 0: |4(0) - 5| = 5 ≤ 2(0) + 9 = 9 ✓

Common Mistakes

Avoid these frequent errors

Forgetting to consider both positive and negative cases
Don't solve only 4x - 5 ≤ 2x + 9 = wrong incomplete solution! Absolute value inequalities always require two cases because the expression inside can be positive or negative. Always solve both cases and find their intersection.

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 split the absolute value into two cases?

Because the expression 4x - 5 inside the absolute value can be either positive or negative! When it's positive, |4x - 5| = 4x - 5. When it's negative, |4x - 5| = -(4x - 5). Both cases must be satisfied simultaneously.

How do I know which case applies to which values of x?

You don't need to worry about that! Just solve both cases separately, then find where they overlap. The final answer is the intersection of both solutions.

What does it mean to combine the solutions?

Take the intersection (overlap) of both cases. From Case 1: $x \leq 7$ . From Case 2: $x \geq -\frac{2}{3}$ . Combined: $-\frac{2}{3} \leq x \leq 7$ .

Can I test my answer by plugging in values?

Absolutely! Try x = 0: $|4(0) - 5| = 5$ and $2(0) + 9 = 9$ . Since 5 ≤ 9, x = 0 works! Try values inside and outside your solution interval.

What if the right side of the inequality is negative?

If the right side is negative, then the inequality has no solution! Absolute values are always non-negative, so |expression| can never be less than a negative number.

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Solve |4x - 5| ≤ 2x + 9: Absolute Value Inequality Challenge

Absolute Value Inequalities with Split Cases

❤️ 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 to split the absolute value into two cases?

How do I know which case applies to which values of x?

What does it mean to combine the solutions?

Can I test my answer by plugging in values?

What if the right side of the inequality is negative?

🌟 Unlock Your Math Potential

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