Math Problem Statement

-7 < 2x - 9 ≤ 1. Solve the inequality.

Solution

Let's solve the given compound inequality step by step:

The inequality is:

7<2x91-7 < 2x - 9 \leq 1

Step 1: Solve the inequality in parts.

First part:

7<2x9-7 < 2x - 9

Add 9 to both sides:

7+9<2x-7 + 9 < 2x 2<2x2 < 2x

Now, divide both sides by 2:

1<x1 < x

Second part:

2x912x - 9 \leq 1

Add 9 to both sides:

2x1+92x \leq 1 + 9 2x102x \leq 10

Now, divide both sides by 2:

x5x \leq 5

Step 2: Combine the results.

From the two parts, we have:

1<x51 < x \leq 5

Step 3: Write the solution in interval notation.

Since 1<x51 < x \leq 5, the interval notation is:

(1,5](1, 5]

Step 4: Graph the solution on the number line.

  • The interval is from 1 to 5.
  • At 1, use an open circle since x>1x > 1.
  • At 5, use a closed circle since x5x \leq 5.

Answer:

  • The correct multiple-choice answer is: 1<x51 < x \leq 5.
  • The interval notation is: (1, 5].

Let me know if you'd like further clarification on any of these steps!


Here are some related questions you might want to explore:

  1. How do you solve compound inequalities with different inequality symbols?
  2. What is the difference between an open and closed interval on a number line?
  3. Can a compound inequality have no solution? In what cases?
  4. How do you graph inequalities on a number line with infinity in the solution?
  5. What is interval notation, and how does it differ from set-builder notation?

Tip: Always remember to handle each inequality in a compound inequality separately before combining the results!

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Math Problem Analysis

Mathematical Concepts

Compound Inequalities
Solving Linear Inequalities

Formulas

Inequality solving process: Isolate the variable
Linear inequality solving

Theorems

Basic inequality principles (addition and division)

Suitable Grade Level

Grades 8-10