Inequalities are used in mathematics whenever you deal with a range of possible values. The inequality could be greater than or less than a certain value, and in some cases inequalities represent ranges that greater/less than or equal to a value. There are some instances where you have more than one constraining value, however; these situations require the use of compound inequalities. A compound inequality is made up of two or more inequalities, connected by "and" or "or" depending on whether you're defining a single range or multiple separate ranges. Solving compound inequalities differs based on whether "and" or "or" is used to link the individual pieces.
TL;DR (Too Long; Didn't Read)
Compound inequalities are solved by isolating your variable on one side of the inequality. If the components are connected by "and," the variable is located between the two constraining values. If the components are connected by "or," the variable inequalities are solved separately.
Compound inequalities connected by "and" look like this: x > 6 and x ≤ 12. In this instance, all valid values of x would be greater than 6, but they would also be less than or equal to 12. The two components of the compound inequality overlap with each other, creating outer bounds for the values of x.
To see how to solve these inequalities, consider the following example: x + 3 < 12 and x – 4 ≥ 0. Solve each portion of the compound inequality to isolate x, giving you x < 9 (by subtracting 3 from each side) and x ≥ 4 (by adding 4 to each side). From this point, arrange the components of the inequality so that x is between the bounds set by the two inequality components. In this case, the solution can be written as 4 ≤ x < 9.
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When compound inequalities are connected by "or", they look like this: x < 5 or x > 10. All of the valid values of x in this example are either less than 5 or greater than 10. Unlike the "and" example above, the inequalities do not overlap.
To solve complex inequalities with "or," consider this example: x – 2 > 7 or x + 1 < 3. As before, solve the two inequalities to isolate x; this gives you x > 9 (by adding 2 to each side) and x < 2 (by subtracting 1 from each side). The solution is written as a union, using ∪ to connect the two inequalities; this looks like (x > 9) ∪ (x < 2).
Graphing Compound Inequalities
When graphing compound inequalities on a line, draw a circle (for > or < inequalities) or dot (for ≥ or ≤ inequalities) at the bound points, or the values you know in the inequalities, to begin your graph. If graphing an "and" inequality, draw a line between the two bound points to complete the graph. If graphing an "or" inequality, draw lines away from the bound points.