Math Problem Statement

Absolute value of x raised to the power of 2 +4 less than 5

Solution

To solve the inequality:

x2+4<5|x|^2 + 4 < 5

Step-by-Step Solution:

  1. Simplify the inequality: Since x2=x2|x|^2 = x^2, we rewrite the inequality as: x2+4<5x^2 + 4 < 5

  2. Isolate x2x^2: Subtract 4 from both sides: x2<1x^2 < 1

  3. Solve for xx: Take the square root of both sides. Remember that when you take the square root of both sides of an inequality, you must consider both the positive and negative roots: 1<x<1-1 < x < 1

Solution:

The solution to the inequality is: x(1,1)x \in (-1, 1)

This means that xx must be between 1-1 and 11 (not including 1-1 and 11) for the inequality to hold.

Would you like a more detailed explanation, or do you have any questions?

Related Questions:

  1. How do absolute values affect inequalities differently from normal expressions?
  2. What changes if the inequality symbol is \leq instead of <<?
  3. How would you solve x3+4<5|x|^3 + 4 < 5?
  4. What if the inequality was x2+4>5x^2 + 4 > 5?
  5. How do you graph solutions to inequalities like this one?

Tip: Always remember to consider both positive and negative solutions when solving inequalities involving squares and square roots.

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Math Problem Analysis

Mathematical Concepts

Absolute Value
Inequalities
Quadratic Expressions

Formulas

|x|^2 = x^2

Theorems

Properties of Absolute Values in Inequalities

Suitable Grade Level

Grades 8-10