Math Problem Statement

Solve for x :

7x−2(x−4)>−2x+1

Solution

Let's solve the inequality step by step:

The inequality is:

7x2(x4)>2x+17x - 2(x - 4) > -2x + 1

Step 1: Distribute the 2-2 on the left side

Distribute 2-2 to both terms inside the parentheses:

7x2(x)+2(4)>2x+17x - 2(x) + 2(4) > -2x + 1

This simplifies to:

7x2x+8>2x+17x - 2x + 8 > -2x + 1

Step 2: Combine like terms

Now combine the terms involving xx on the left side:

5x+8>2x+15x + 8 > -2x + 1

Step 3: Move the xx-terms to one side

To eliminate 2x-2x from the right side, add 2x2x to both sides:

5x+2x+8>15x + 2x + 8 > 1

This simplifies to:

7x+8>17x + 8 > 1

Step 4: Move the constant term to the other side

Subtract 8 from both sides:

7x>187x > 1 - 8

7x>77x > -7

Step 5: Solve for xx

Now divide both sides by 7:

x>77x > \frac{-7}{7}

x>1x > -1

Final Answer:

The solution to the inequality is:

x>1x > -1


Would you like any further explanation on this solution? Or have any additional questions?

Here are some follow-up questions to explore:

  1. How do you handle inequalities involving absolute values?
  2. What happens if both sides of an inequality are multiplied by a negative number?
  3. How can we graph the solution x>1x > -1 on a number line?
  4. What are the key differences between solving an equation and solving an inequality?
  5. What if the inequality was 7x2(x4)2x+17x - 2(x - 4) \geq -2x + 1? How would that change the solution?

Tip: Always pay attention when multiplying or dividing by negative numbers in inequalities; it flips the inequality sign!

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Math Problem Analysis

Mathematical Concepts

Algebra
Inequalities
Linear Equations

Formulas

Distributive Property
Solving linear inequalities

Theorems

Basic rules for solving inequalities

Suitable Grade Level

Grades 7-9