Math Problem Statement

Solve the compound inequality: -10 < 4x - 2 ≤ 10 and graph the solution on the number line.

Solution

To solve the compound inequality:

10<4x210-10 < 4x - 2 \leq 10

we can break it down into two parts:

Step 1: Solve the left inequality

10<4x2-10 < 4x - 2

Add 2 to both sides: 10+2<4x-10 + 2 < 4x 8<4x-8 < 4x

Then, divide by 4: 2<x-2 < x

This gives us: x>2x > -2

Step 2: Solve the right inequality

4x2104x - 2 \leq 10

Add 2 to both sides: 4x124x \leq 12

Then, divide by 4: x3x \leq 3

Combine the Results

The solution for the compound inequality is: 2<x3-2 < x \leq 3

Graphing the Solution

On the number line:

  • Draw an open circle at x=2x = -2 (since x>2x > -2 does not include -2).
  • Draw a closed circle at x=3x = 3 (since x3x \leq 3 includes 3).
  • Shade the region between x=2x = -2 and x=3x = 3.

Would you like more details on each step, or have any questions about this solution?


Here are 5 related questions to expand your understanding:

  1. How would the solution change if the inequality were 104x2<10-10 \leq 4x - 2 < 10?
  2. How do you solve a compound inequality with "or" instead of "and"?
  3. What are the steps to graph a strict inequality versus a non-strict inequality on a number line?
  4. How can we verify that our solution to a compound inequality is correct?
  5. How does dividing or multiplying by a negative number affect an inequality?

Tip: Always pay attention to the inequality signs, especially when dividing by negative numbers, as they reverse the direction of the inequality.

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Math Problem Analysis

Mathematical Concepts

Algebra
Compound Inequalities
Inequality Solving

Formulas

Solving compound inequalities by isolating the variable

Theorems

Inequality properties for addition, subtraction, multiplication, and division

Suitable Grade Level

Grades 7-10