Solve the Absolute Value Equation: |3x - 5| = 4

Q: Why does an absolute value equation have two answers?

Because absolute value measures distance from zero, both positive and negative numbers can have the same absolute value. For example, both 4 and -4 are distance 4 from zero, so $ |4| = |-4| = 4 $.

Q: How do I know when to use positive and negative cases?

Always create both cases! If $ |expression| = number $, then expression = number OR expression = -number. This gives you two separate equations to solve.

Q: Can an absolute value equation have no solutions?

Yes! If the right side is negative, there are no solutions because absolute values are never negative. For example, $ |x - 2| = -3 $ has no solutions.

Q: Do I need to check my answers?

Absolutely! Substitute each solution back into the original equation. Both $ x = 3 $ and $ x = \frac{1}{3} $ should make $ |3x - 5| = 4 $ true.

Absolute Value Equations with Two Solutions

$|3x - 5| = 4$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$|3x - 5| = 4$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Consider the positive scenario: $3x - 5 = 4$
Step 2: Consider the negative scenario: $3x - 5 = -4$
Step 3: Solve each equation separately
Step 4: Identify the correct answer from the given choices

Let's work through each step:

Step 1: Solve $3x - 5 = 4$ .
Add 5 to both sides: $3x - 5 + 5 = 4 + 5$
So, $3x = 9$ .
Divide both sides by 3: $x = \frac{9}{3}$ , thus $x = 3$ .

Step 2: Solve $3x - 5 = -4$ .
Add 5 to both sides: $3x - 5 + 5 = -4 + 5$
So, $3x = 1$ .
Divide both sides by 3: $x = \frac{1}{3}$ .

Step 3: We have two solutions for $x$ , namely $x = 3$ and $x = \frac{1}{3}$ .

Step 4: Comparing this with the given choices, the correct answer is "Answers b + c":

$x = 3$ corresponds to choice b.
$x = \frac{1}{3}$ corresponds to choice c.

Therefore, the solution to the problem is Answers b + c.

Final Answer

Answers b + c

Key Points to Remember

Essential concepts to master this topic

Definition: Absolute value equations create two separate linear equations
Method: Set $3x - 5 = 4$ and $3x - 5 = -4$
Verification: Check both $|3(3) - 5| = 4$ and $|3(\frac{1}{3}) - 5| = 4$ ✓

Common Mistakes

Avoid these frequent errors

Only solving one case of the absolute value equation
Don't just solve 3x - 5 = 4 and stop = missing half the solutions! Absolute value means the expression inside could be positive OR negative. Always solve both 3x - 5 = 4 AND 3x - 5 = -4 to find all solutions.

Practice Quiz

Test your knowledge with interactive questions

$\left|x\right|=3$

FAQ

Everything you need to know about this question

Why does an absolute value equation have two answers?

Because absolute value measures distance from zero, both positive and negative numbers can have the same absolute value. For example, both 4 and -4 are distance 4 from zero, so $|4| = |-4| = 4$ .

How do I know when to use positive and negative cases?

Always create both cases! If $|expression| = number$ , then expression = number OR expression = -number. This gives you two separate equations to solve.

What if I get the same answer from both cases?

That's possible but rare! It usually means the expression inside the absolute value bars equals zero at your solution. Always check both cases even if you think they might be the same.

Can an absolute value equation have no solutions?

Yes! If the right side is negative, there are no solutions because absolute values are never negative. For example, $|x - 2| = -3$ has no solutions.

Do I need to check my answers?

Absolutely! Substitute each solution back into the original equation. Both $x = 3$ and $x = \frac{1}{3}$ should make $|3x - 5| = 4$ true.

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