Math Problem Statement

x26<5\left\vert x^2-6\right\vert<5

Solution

To solve the inequality x26<5|x^2 - 6| < 5, we follow these steps:

Step 1: Rewrite the inequality without the absolute value

The expression x26<5|x^2 - 6| < 5 implies: 5<x26<5-5 < x^2 - 6 < 5

Step 2: Solve the compound inequality

Break this into two parts: 5<x26andx26<5-5 < x^2 - 6 \quad \text{and} \quad x^2 - 6 < 5

Part 1: Solve 5<x26-5 < x^2 - 6

Add 6 to both sides: 1<x21 < x^2 This simplifies to: x2>1x^2 > 1

Part 2: Solve x26<5x^2 - 6 < 5

Add 6 to both sides: x2<11x^2 < 11

Step 3: Combine the results

We now combine x2>1x^2 > 1 and x2<11x^2 < 11. Together, this means: 1<x2<111 < x^2 < 11

Step 4: Solve for xx

The inequality 1<x2<111 < x^2 < 11 can be solved by taking square roots:

  1. From x2>1x^2 > 1: x>1x > 1 or x<1x < -1.
  2. From x2<11x^2 < 11: 11<x<11-\sqrt{11} < x < \sqrt{11}.

Combining these, we have: x(11,1)(1,11)x \in (-\sqrt{11}, -1) \cup (1, \sqrt{11})

Final Solution:

The solution to the inequality is: x(11,1)(1,11)x \in (-\sqrt{11}, -1) \cup (1, \sqrt{11})


Would you like a detailed graph or further clarification?


5 Related Questions:

  1. How do you solve absolute value inequalities in general?
  2. What happens when x26>5|x^2 - 6| > 5?
  3. How does squaring affect the inequality when solving x2<kx^2 < k?
  4. Can you explain the difference between strict and non-strict inequalities?
  5. What is the graphical representation of x26<5|x^2 - 6| < 5?

Tip:

When dealing with absolute value inequalities, always remember to split the inequality into two cases—one positive and one negative!

Ask a new question for Free

By Image

Drop file here or Click Here to upload

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

-

Suitable Grade Level

Grades 9-12