Solving Inequalities

Inequalities are written using the symbols <, >, ≤, and ≥. For example, x < 4 means that “x is less than 4”.
To solve an inequality, use the same steps as you would to solve an equation. However, when both sides are multiplied or divided by a negative number, you must reverse the inequality symbol.
Example: -2x < 4. To isolate x, divide both sides by -2, and reverse the inequality symbol: x > -2.
Combine inequalities using “and” if there is an overlapping range and “or” if the ranges are separate.

Working with Quadratic Inequalities

Quadratic inequalities have an x² term. They are solved by finding the “roots” or “zeros” of the inequality (the x-values where y equals zero).
Begin by setting the quadratic inequality to 0, then factor or use the quadratic formula to find the zero points.
To determine what values of x make the inequality true, test a value on either side of a root in the original inequality.
Once you have determined which sections are true or false, represent the solutions on a number line and in interval notation.

Absolute value inequalities contain expressions within absolute value brackets ( ).
An equation such as x < a has a solution of all values between -a and a: -a < x < a.
An equation such as x > a has a solution of all values less than -a and greater than a: x < -a or x > a.
To solve more complicated absolute value inequalities, isolate the absolute value term, then solve as a compound inequality as noted above.