Question

Inequalities portfolio

Answer 1

Inequalities portfolio

- Problem 1:
Solve the inequality: 2x + 5 > 10

Step 1: Subtract 5 from both sides of the inequality: 2x > 5
Step 2: Next, divide both sides of the inequality by 2: x > 2.5
The solution to the inequality is x > 2.5

- Problem 2:
Solve the inequality: 3(2x - 4) ≤ 9x + 7

Step 1: Distribute the 3 on the left side: 6x - 12 ≤ 9x + 7
Step 2: Subtract 6x from both sides of the inequality: -12 ≤ 3x + 7
Step 3: Subtract 7 from both sides of the inequality: -19 ≤ 3x
Step 4: Divide both sides of the inequality by 3: -6.33 ≤ x
The solution to the inequality is x ≥ -6.33

- Problem 3:
Solve the compound inequality: -3 < 2x + 5 ≤ 7

Step 1: Subtract 5 from all parts of the compound inequality: -8 < 2x ≤ 2
Step 2: Divide all parts of the compound inequality by 2: -4 < x ≤ 1
The solution to the compound inequality is -4 < x ≤ 1

- Problem 4:
Solve the absolute value inequality: |3x - 2| < 5

Case 1: 3x - 2 > 0
Solve the inequality: 3x - 2 < 5
Add 2 to both sides: 3x < 7
Divide both sides by 3: x < 7/3

Case 2: 3x - 2 < 0
Solve the inequality: -(3x - 2) < 5
Distribute the negative sign: -3x + 2 < 5
Subtract 2 from both sides: -3x < 3
Divide both sides by -3: x > -1

Combine the solutions from both cases: -1 < x < 7/3
The solution to the absolute value inequality is -1 < x < 7/3.