Solve the Absolute Value Equation: |z - 7| = 3

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

Absolute value measures distance from zero, and two different numbers can be the same distance away! For example, both 3 and -3 are exactly 3 units from zero.

Q: How do I know which sign to use for the second case?

Always use the opposite of the number on the right side. If you have $ |z - 7| = 3 $, your two cases are $ z - 7 = 3 $ and $ z - 7 = -3 $.

Q: Can absolute value equations have no solution?

Yes! If the right side is negative, there's no solution because absolute values are never negative. For example, $ |z - 7| = -2 $ has no solution.

Q: Do I always add or subtract the same number to both sides?

Yes! Whatever operation you need to isolate the variable, do it to both sides of each equation. In this problem, we added 7 to both sides of each case.

Absolute Value Equations with Two Solutions

Find the value of $z$ such that $|z - 7| = 3$ .

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Find the value of $z$ such that $|z - 7| = 3$ .

Step-by-step solution

Given the equation $|z - 7| = 3$ , consider the two cases for the absolute value:

Case 1: $z - 7 = 3$
Add 7: $z = 10$ .
Case 2: $z - 7 = -3$
Add 7: $z = 4$ .

Thus, $z$ is $10 \text{ or } 4$ .

Final Answer

$z = 10 \text{ or } z = 4$

Key Points to Remember

Essential concepts to master this topic

Definition: |expression| = number creates two separate linear equations
Method: Set z - 7 = 3 and z - 7 = -3
Check: Substitute both answers: |10 - 7| = 3 and |4 - 7| = 3 ✓

Common Mistakes

Avoid these frequent errors

Forgetting the negative case
Don't only solve z - 7 = 3 to get z = 10! This misses half the solution because absolute value creates distance in both directions. Always solve both z - 7 = 3 AND z - 7 = -3 to find all solutions.

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 does an absolute value equation have two answers?

Absolute value measures distance from zero, and two different numbers can be the same distance away! For example, both 3 and -3 are exactly 3 units from zero.

How do I know which sign to use for the second case?

Always use the opposite of the number on the right side. If you have $|z - 7| = 3$ , your two cases are $z - 7 = 3$ and $z - 7 = -3$ .

What if I get the same answer for both cases?

This can happen! If the expression inside the absolute value bars equals zero, you'll get one repeated solution. That's still correct - just write it once.

Can absolute value equations have no solution?

Yes! If the right side is negative, there's no solution because absolute values are never negative. For example, $|z - 7| = -2$ has no solution.

Do I always add or subtract the same number to both sides?

Yes! Whatever operation you need to isolate the variable, do it to both sides of each equation. In this problem, we added 7 to both sides of each case.

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