Solve the Absolute Value Equation: |x + 1| = 5

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

Think of absolute value as distance! If $ |x + 1| = 5 $, then x + 1 is 5 units away from zero. This can happen in two ways: $ x + 1 = 5 $ or $ x + 1 = -5 $.

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

It doesn't matter! You need to solve both cases anyway. Some students prefer the positive case first, but either order gives you the same final answers.

Q: Can absolute value equal a negative number?

No! Absolute value represents distance, which is always positive or zero. If you see $ |x + 1| = -3 $, there are no solutions.

Q: How do I check if my solutions are correct?

Substitute each solution back into the original equation. For $ x = 4 $: $ |4 + 1| = |5| = 5 $ ✓. For $ x = -6 $: $ |-6 + 1| = |-5| = 5 $ ✓

Absolute Value Equations with Two Solutions

$\left|x+1\right|=5$

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$\left|x+1\right|=5$

Step-by-step solution

To solve the absolute value equation $|x + 1| = 5$ , we follow these steps:

Step 1: Understand that the equation $|x + 1| = 5$ means the expression inside the absolute value, $x + 1$ , is equal to 5 or -5.
Step 2: Set up two separate equations:

Equation 1: $x + 1 = 5$
Equation 2: $x + 1 = -5$

Step 3: Solve each equation individually.

For Equation 1: Subtract 1 from both sides to get $x = 5 - 1 = 4$ .
For Equation 2: Subtract 1 from both sides to get $x = -5 - 1 = -6$ .

Thus, the solutions to the equation $|x + 1| = 5$ are $x = 4$ and $x = -6$ .

Therefore, the correct answer, considering the choices provided, is Answer a + b, which corresponds to choices 1 and 2.

Final Answer

Answers a + b

Key Points to Remember

Essential concepts to master this topic

Definition: $|x + 1| = 5$ means distance from -1 is 5
Technique: Set up two cases: $x + 1 = 5$ and $x + 1 = -5$
Check: Both solutions: $|4 + 1| = 5$ and $|-6 + 1| = 5$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting the negative case
Don't just solve $x + 1 = 5$ and stop = you miss half the solutions! Absolute value creates distance, which can come from positive OR negative direction. Always set up both $x + 1 = 5$ AND $x + 1 = -5$ .

Practice Quiz

Test your knowledge with interactive questions

$\left|x\right|=5$

FAQ

Everything you need to know about this question

Why does an absolute value equation have two solutions?

Think of absolute value as distance! If $|x + 1| = 5$ , then x + 1 is 5 units away from zero. This can happen in two ways: $x + 1 = 5$ or $x + 1 = -5$ .

How do I know which equation to solve first?

It doesn't matter! You need to solve both cases anyway. Some students prefer the positive case first, but either order gives you the same final answers.

What if I get the same answer from both cases?

That's possible! Sometimes absolute value equations have just one solution. Always solve both cases completely - if they give the same result, that's your single answer.

Can absolute value equal a negative number?

No! Absolute value represents distance, which is always positive or zero. If you see $|x + 1| = -3$ , there are no solutions.

How do I check if my solutions are correct?

Substitute each solution back into the original equation. For $x = 4$ : $|4 + 1| = |5| = 5$ ✓. For $x = -6$ : $|-6 + 1| = |-5| = 5$ ✓

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Absolute Value and Inequality questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime