Math Problem Statement

4x-4<8 \quad \maroonC{\text{ AND}} \quad 9x+5>23

Solution

Let's solve the two inequalities step by step:

First inequality:

4x4<84x - 4 < 8

  1. Add 4 to both sides: 4x<8+44x < 8 + 4 4x<124x < 12

  2. Divide by 4: x<124x < \frac{12}{4} x<3x < 3

Second inequality:

9x+5>239x + 5 > 23

  1. Subtract 5 from both sides: 9x>2359x > 23 - 5 9x>189x > 18

  2. Divide by 9: x>189x > \frac{18}{9} x>2x > 2

Combining the two inequalities:

We have: x<3ANDx>2x < 3 \quad \text{AND} \quad x > 2

Thus, the solution is: 2<x<32 < x < 3

Would you like more details on this?

Here are 5 related questions for further exploration:

  1. How would the solution change if we had 4x484x - 4 \leq 8 and 9x+5239x + 5 \geq 23?
  2. Can you represent the solution 2<x<32 < x < 3 on a number line?
  3. How do the rules for solving inequalities differ from solving equations?
  4. What would happen if one inequality had no solution?
  5. Can inequalities be solved using graphing techniques?

Tip: Always remember to flip the inequality sign when multiplying or dividing by a negative number!

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Math Problem Analysis

Mathematical Concepts

Algebra
Inequalities
Linear Equations

Formulas

Linear inequality manipulation: ax + b < c
Linear inequality manipulation: ax + b > c

Theorems

Properties of inequalities
Addition and subtraction property of inequalities
Division and multiplication property of inequalities

Suitable Grade Level

Grades 7-9