- Submitted By: userq45
- Date Submitted: 11/01/2014 7:36 PM
- Category: Miscellaneous
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- Page: 6

Solving Inequalities

An inequality is the result of replacing the = sign in an equation with , ≤, or ≥. For

example, 3x – 2 < 7 is a linear inequality. We call it “linear” because if the < were

replaced with an = sign, it would be a linear equation. Inequalities involving polynomials

of degree 2 or more, like 2x3 – x > 0, are referred to as polynomial inequalities, or

quadratic inequalities if the degree is exactly 2. Inequalities involving rational

expressions are called rational inequalities. Some often used inequalities also involve

absolute value expressions.

Solving Inequalities: A Summary

In a nutshell, solving inequalities is about one thing: sign changes. Find all the points at

which there are sign changes - we call these points critical values. Then determine

which, if any, of the intervals bounded by these critical values result in a solution. The

solution to the inequality will consist of the set of all points contained by the solution

intervals.

Method To Solve Linear, Polynomial, or Absolute Value Inequalities:

1. Move all terms to one side of the inequality sign by applying the Addition,

Subtraction, Multiplication, and Division Properties of Inequalities. You should have

only zero on one side of the inequality sign.

2. Solve the associated equation using an appropriate method. This solution or

solutions will make up the set of critical values. At these values, sign changes occur

in the inequality.

3. Plot the critical values on a number line. Use closed circles • for ≤ and ≥

inequalities, and use open circles ο for < and > inequalities.

4. Test each interval defined by the critical values. If an interval satisfies the

inequality, then it is part of the solution. If it does not satisfy the inequality, then it is

not part of the solution.

Example: Solve 3x + 5(x + 1) ≤ 4x – 1 and graph the solution

3x + 5(x + 1) ≤ 4x – 1

3x + 5x + 5 ≤ 4x – 1

8x + 5 ≤ 4x – 1

4x + 6 ≤ 0

Now, solve 4x+6 = 0

4x = -6

x = - 6/4 = -3/2...