Solving inequalities using multiplication and division

Answer 1

To solve inequalities using multiplication and division, follow these steps:

1. If the inequality has a variable on both sides of the inequality sign, move all the terms containing the variable to one side of the inequality by adding or subtracting as needed.

2. Identify whether you need to multiply or divide both sides of the inequality by a positive or negative number.

- If you multiply or divide both sides by a positive number, the inequality sign remains the same.
- If you multiply or divide both sides by a negative number, the inequality sign flips.

3. Perform the multiplication or division operation on both sides of the inequality.

4. Simplify the inequality by combining like terms and obtaining a solution.

5. If necessary, express the solution as an interval or set notation.

Note: Whenever you multiply or divide both sides of an inequality by a negative number, remember to flip the inequality sign.

Example 1: Solve the inequality 3x - 6 < 9.

Step 1: Move all terms involving x to the left side: 3x < 9 + 6.
Simplified: 3x < 15.

Step 2: Since the coefficient of x is positive, we can proceed with multiplication or division without flipping the inequality sign.

Step 3: Divide both sides by 3 to isolate x: x < 5.

Step 4: Simplify the inequality: x < 5.

Step 5: The solution can be written in interval notation as (-∞, 5).

Example 2: Solve the inequality -2(4 - x) ≥ 10.

Step 1: Distribute the negative sign inside the parentheses: -8 + 2x ≥ 10.

Step 2: Since the coefficient of x is positive, we can proceed with multiplication or division without flipping the inequality sign.

Step 3: Add 8 to both sides to isolate 2x: 2x ≥ 18.

Step 4: Divide both sides by 2 to solve for x: x ≥ 9.

Step 5: The solution can be written in interval notation as [9, +∞).