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...