Math Problem Statement

Solve the equations: (a) |a + 5| = 3, (b) |a + 5| = 0, (c) |a + 5| = -2

Solution

Let's solve each of the given equations one by one.

(a) a+5=3\left| a + 5 \right| = 3

For an absolute value equation like a+5=3\left| a + 5 \right| = 3, we have two cases to solve:

  1. a+5=3a + 5 = 3 a=35=2a = 3 - 5 = -2

  2. a+5=3a + 5 = -3 a=35=8a = -3 - 5 = -8

So, the solution set for part (a) is a=2a = -2 and a=8a = -8.

(b) a+5=0\left| a + 5 \right| = 0

The absolute value of any expression is zero only if the expression itself is zero. Therefore:

a+5=0a + 5 = 0 a=5a = -5

So, the solution set for part (b) is a=5a = -5.

(c) a+5=2\left| a + 5 \right| = -2

An absolute value cannot be negative, so there are no solutions for this equation.


Summary of Solutions:

  • (a) a=2,8a = -2, -8
  • (b) a=5a = -5
  • (c) No solution

Would you like further details on any step? Here are five related questions to expand on this:

  1. What are the properties of absolute value equations?
  2. How can absolute value inequalities be solved?
  3. Can absolute value equations have no solutions, and why?
  4. How do you graph absolute value functions?
  5. What is the geometric interpretation of an absolute value equation?

Tip: When solving absolute value equations, always remember to check for both the positive and negative cases!

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Math Problem Analysis

Mathematical Concepts

Absolute Value
Linear Equations

Formulas

|x| = y implies x = y or x = -y

Theorems

The definition of absolute value
No solution for negative absolute values

Suitable Grade Level

Grades 6-8