Solve the Absolute Value Equation: |-5x + 2| = 13

Q: Why do I need two equations for one absolute value problem?

Because absolute value measures distance from zero, and two different numbers can be the same distance away! For example, both 3 and -3 are distance 3 from zero, so |x| = 3 has solutions x = 3 and x = -3.

Q: How do I know which equation to solve first?

It doesn't matter! You can solve $ -5x + 2 = 13 $ first or $ -5x + 2 = -13 $ first. Just make sure you solve both equations completely.

Q: Can absolute value equations have no solutions?

Yes! If you have something like $ |x + 5| = -3 $, there's no solution because absolute values are never negative. Always check if your equation makes sense.

Q: How do I check my answers are correct?

Substitute each solution back into the original equation. For x = 3: $ |-5(3) + 2| = |-15 + 2| = |-13| = 13 ✓ $. Do the same check for x = -2.2!

Absolute Value Equations with Linear Expressions

$\left|-5x + 2\right|=13$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$\left|-5x + 2\right|=13$

Step-by-step solution

To solve $\left|-5x + 2\right|=13$ , address the two absolute value cases:

1) $-5x + 2=13$ :

Subtract $2$ from both sides to get $-5x=11$ .

Divide by $-5$ to find $x=-2.2$ .

2) $-5x + 2=-13$ :

Subtract $2$ from both sides to get $-5x=-15$ .

Divide by $-5$ for $x=3$ .

The solutions are $x=3$ and $x=-2.2$ .

Final Answer

$x=3$ , $x=-2.2$

Key Points to Remember

Essential concepts to master this topic

Rule: Absolute value equals positive number creates two separate equations
Technique: Set -5x + 2 = 13 and -5x + 2 = -13
Check: Substitute both solutions: |-5(3) + 2| = 13 and |-5(-2.2) + 2| = 13 ✓

Common Mistakes

Avoid these frequent errors

Forgetting the negative case when solving absolute value equations
Don't solve only -5x + 2 = 13 and stop there = missing half the solutions! The absolute value creates two possibilities since |A| = B means A = B OR A = -B. Always set up both equations: the expression equals the positive value AND the negative value.

Practice Quiz

Test your knowledge with interactive questions

$\left|-x\right|=10$

FAQ

Everything you need to know about this question

Why do I need two equations for one absolute value problem?

Because absolute value measures distance from zero, and two different numbers can be the same distance away! For example, both 3 and -3 are distance 3 from zero, so |x| = 3 has solutions x = 3 and x = -3.

How do I know which equation to solve first?

It doesn't matter! You can solve $-5x + 2 = 13$ first or $-5x + 2 = -13$ first. Just make sure you solve both equations completely.

What if I get the same answer from both equations?

That's possible! Sometimes absolute value equations have only one solution. This happens when the expression inside equals zero at the boundary. Always check both cases anyway.

Can absolute value equations have no solutions?

Yes! If you have something like $|x + 5| = -3$ , there's no solution because absolute values are never negative. Always check if your equation makes sense.

How do I check my answers are correct?

Substitute each solution back into the original equation. For x = 3: $|-5(3) + 2| = |-15 + 2| = |-13| = 13 ✓$ . Do the same check for x = -2.2!

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