Math Problem Statement
Solution
To solve the inequality , we follow these steps:
Step 1: Rewrite the inequality without the absolute value
The expression implies:
Step 2: Solve the compound inequality
Break this into two parts:
Part 1: Solve
Add 6 to both sides: This simplifies to:
Part 2: Solve
Add 6 to both sides:
Step 3: Combine the results
We now combine and . Together, this means:
Step 4: Solve for
The inequality can be solved by taking square roots:
- From : or .
- From : .
Combining these, we have:
Final Solution:
The solution to the inequality is:
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5 Related Questions:
- How do you solve absolute value inequalities in general?
- What happens when ?
- How does squaring affect the inequality when solving ?
- Can you explain the difference between strict and non-strict inequalities?
- What is the graphical representation of ?
Tip:
When dealing with absolute value inequalities, always remember to split the inequality into two cases—one positive and one negative!
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Math Problem Analysis
Mathematical Concepts
Algebra
Inequalities
Absolute Value
Formulas
a - b < c implies a < b + c and a > b - c
Properties of Absolute Value
Properties of Inequalities
Theorems
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Suitable Grade Level
Grades 9-12