How to Solve an Inequality Equation?

Solving an inequality equation involves finding the range of values that satisfy the inequality. Here's a breakdown of the process:

1. Simplify Both Sides:

• Combine like terms on each side of the inequality.
• Get rid of parentheses by distributing.

2. Isolate the Variable Term:

• Use addition or subtraction to move all terms containing the variable to one side of the inequality and all constant terms to the other side. Remember to perform the same operation on both sides to maintain the inequality.

3. Solve for the Variable:

• Multiply or divide both sides of the inequality by the coefficient of the variable.
  • Crucially: If you multiply or divide by a negative number, you must reverse the direction of the inequality sign.

4. Express the Solution:

• The solution will typically be a range of values. Express this range using inequality notation (e.g., x < 5, x ≥ -2) or interval notation (e.g., (-∞, 5), [-2, ∞)).
• You can also represent the solution graphically on a number line.

Example 1: Simple Inequality

Solve: 3x + 2 < 8

1. Subtract 2 from both sides: 3x < 6
2. Divide both sides by 3: x < 2

Solution: x < 2 or (-∞, 2)

Example 2: Inequality with a Negative Coefficient

Solve: -2x + 5 ≥ 11

1. Subtract 5 from both sides: -2x ≥ 6
2. Divide both sides by -2 (and reverse the inequality sign): x ≤ -3

Solution: x ≤ -3 or (-∞, -3]

Example 3: Inequality with Fractions

Solve: (1/2)x + 1 < (3/4)x + 2

1. Multiply every term on both sides by the LCD (Least Common Denominator) to eliminate fractions. In this case, the LCD of 2 and 4 is 4.

   4 (1/2)x + 4 1 < 4 (3/4)x + 4 2

2. Simplify: 2x + 4 < 3x + 8

3. Subtract 2x from both sides: 4 < x + 8

4. Subtract 8 from both sides: -4 < x

5. Rewrite: x > -4

Solution: x > -4 or (-4, ∞)

Key Considerations:

• Reversing the Inequality Sign: Always remember to reverse the inequality sign when multiplying or dividing by a negative number.
• Interval Notation: Understand how to represent solutions using interval notation. Parentheses indicate that the endpoint is not included, while brackets indicate that it is included.
• Graphing: Visualizing the solution on a number line can help you understand the range of values that satisfy the inequality.
• Compound Inequalities: Some problems might involve compound inequalities, such as "and" or "or" inequalities. Solve each part separately and then combine the solutions.