Solve the Absolute Value Equation: Finding x in |3x + 1| = 4

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

Because absolute value measures distance from zero! When $|something| = 4$, that 'something' could be 4 units away in either direction: +4 or -4.

Q: How do I know which answer choices are correct?

Solve both cases completely: $x = 1$ and $x = -\frac{5}{3}$. Then check which of these values appear in your answer choices. Both solutions are mathematically correct!

Q: Can an absolute value equation have no solutions?

Yes! If the right side is negative (like $|x + 2| = -3$), there are no solutions because absolute value is never negative.

Q: Do I always get exactly two solutions?

Not always. You get two solutions when the right side is positive, one solution when it equals zero, and no solutions when it's negative.

Absolute Value Equations with Two Solutions

Solve for $x$ : $\left|3x + 1\right| = 4$

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Step-by-step written solution

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Understand the problem

Solve for $x$ : $\left|3x + 1\right| = 4$

Step-by-step solution

To solve the absolute value equation $\left|3x + 1\right| = 4$ , we set up two separate equations because absolute value represents the distance from zero, meaning the expression inside can be equal to 4 or -4.

First equation: $3x + 1 = 4$
Subtract 1 from both sides: $3x = 3$
Divide both sides by 3: $x = 1$

Second equation: $3x + 1 = -4$
Subtract 1 from both sides: $3x = -5$
Divide both sides by 3: $x = -\frac{5}{3}$

Therefore, the solutions are $x = 1$ and $x = -\frac{5}{3}$ , but only the negative solution satisfies the original setup with a valid subtraction and division sequence again as per the equation.

Final Answer

$x = \frac{5}{3}$

Key Points to Remember

Essential concepts to master this topic

Rule: Absolute value equation creates two separate linear equations
Technique: Set 3x + 1 = 4 and 3x + 1 = -4
Check: Substitute both solutions: |3(1) + 1| = 4 and |3(-5/3) + 1| = 4 ✓

Common Mistakes

Avoid these frequent errors

Forgetting the negative case
Don't solve only 3x + 1 = 4 and ignore the negative case = missing half the solutions! Absolute value means distance from zero, so the expression inside can equal both +4 and -4. Always set up both equations: 3x + 1 = 4 AND 3x + 1 = -4.

Practice Quiz

Test your knowledge with interactive questions

Determine the absolute value of the following number:

$\left|18\right|=$

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! When $|something| = 4$ , that 'something' could be 4 units away in either direction: +4 or -4.

How do I know which answer choices are correct?

Solve both cases completely: $x = 1$ and $x = -\frac{5}{3}$ . Then check which of these values appear in your answer choices. Both solutions are mathematically correct!

Can an absolute value equation have no solutions?

Yes! If the right side is negative (like $|x + 2| = -3$ ), there are no solutions because absolute value is never negative.

What if I get the same answer from both equations?

That's possible! It means the expression inside the absolute value bars equals zero. For example, $|x - 5| = 0$ gives only $x = 5$ .

Do I always get exactly two solutions?

Not always. You get two solutions when the right side is positive, one solution when it equals zero, and no solutions when it's negative.

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