Solve the Absolute Value Equation: Find y in |y - 3| = 7

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

Because absolute value measures distance, and two different numbers can be the same distance from a point! For $ |y - 3| = 7 $, both 10 and -4 are exactly 7 units away from 3 on the number line.

Q: How do I set up the two cases?

If you have $ |expression| = number $, then:Case 1: expression = numberCase 2: expression = -numberFor our problem: y - 3 = 7 and y - 3 = -7

Q: Can an absolute value equation have no solution?

Yes! If the right side is negative (like $ |y - 3| = -5 $), there's no solution because absolute values are never negative.

Q: How do I check my answers?

Substitute each solution back into the original equation. For y = 10: $ |10 - 3| = |7| = 7 $ ✓For y = -4: $ |-4 - 3| = |-7| = 7 $ ✓

Absolute Value Equations with Two-Case Solutions

$\left| y - 3 \right| = 7$

❤️ Continue Your Math Journey!

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

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$\left| y - 3 \right| = 7$

Step-by-step solution

To solve $\left| y - 3 \right| = 7$ , we need to consider the two possible cases for the absolute value equation.

1. $y - 3 = 7$ leads to $y = 10$

2. $y - 3 = -7$ leads to $y = -4$

Thus, the solutions are $y = 10$ and $y = -4$ .

Final Answer

$y = 10$ , $y = -4$

Key Points to Remember

Essential concepts to master this topic

Rule: Absolute value equations create two separate cases to solve
Technique: For |y - 3| = 7, solve y - 3 = 7 and y - 3 = -7
Check: Substitute both answers: |10 - 3| = 7 and |-4 - 3| = 7 ✓

Common Mistakes

Avoid these frequent errors

Solving only the positive case
Don't solve just y - 3 = 7 to get y = 10 and stop there! This gives you only half the solution set. Always solve both the positive case (y - 3 = 7) and negative case (y - 3 = -7) 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, and two different numbers can be the same distance from a point! For $|y - 3| = 7$ , both 10 and -4 are exactly 7 units away from 3 on the number line.

How do I set up the two cases?

If you have $|expression| = number$ , then:
Case 1: expression = number
Case 2: expression = -number
For our problem: y - 3 = 7 and y - 3 = -7

What if I get the same answer for both cases?

That's possible! Sometimes absolute value equations have only one solution. This happens when the expression inside equals zero. Always solve both cases, but don't worry if they give the same result.

Can an absolute value equation have no solution?

Yes! If the right side is negative (like $|y - 3| = -5$ ), there's no solution because absolute values are never negative.

How do I check my answers?

Substitute each solution back into the original equation. For y = 10: $|10 - 3| = |7| = 7$ ✓
For y = -4: $|-4 - 3| = |-7| = 7$ ✓

🌟 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