A compound inequality is an inequality that combines two simple inequalities. This article provides a review of how to graph and solve compound inequalities.

What is a compound inequality?

A compound inequality is an inequality that combines two simple inequalities. Let's take a look at some examples.

Example with "OR"

x, is less than, 3, space, start color purpleD, space, O, R, space, end color purpleD, space, x, is greater than, 5
So, for example, the numbers 0 and 6 are both solutions of the compound inequality, but the number 4 is not a solution.

Example with "AND"

x, is greater than, 0, space, start color goldD, space, A, N, D, space, end color goldD, space, x, is less than, 4
This compound inequality is true for values that are both greater than zero and less than four. Graphically, we represent it like this:
So, in this case, 2 is a solution of the compound inequality, but 5 is not because it only satisfies one of the inequalities, not both.
Note: If we wanted to, we could write this compound inequality more simply like this:
0, is less than, x, is less than, 4

Solving compound inequalities

Example with "OR"

Solve for x.
2, x, plus, 3, is greater than or equal to, 7, space, start color purpleD, space, O, R, space, end color purpleD, space, 2, x, plus, 9, is greater than, 11
Solving the first inequality for x, we get:
2x+372x4x2\begin{aligned} 2x+3 &\geq 7 \\\\ 2x &\geq 4 \\\\ x &\geq 2 \end{aligned}
Solving the second inequality for x, we get:
2x+9>112x>2x>1\begin{aligned} 2x+9&>11 \\\\ 2x&>2\\\\ x&>1 \end{aligned}
Graphically, we get:
So our compound inequality can be expressed as the simple inequality:
x, is greater than, 1
Want to learn more about compound inequalities that use OR statements? Check out this video.

Example with "AND"

Solve for x.
4, x, minus, 39, is greater than, minus, 43, space, start color goldD, space, A, N, D, end color goldD, space, 8, x, plus, 31, is less than, 23
Solving the first inequality for x, we get:
4x39>434x>4x>1\begin{aligned}4x-39&> -43 \\\\ 4x &> -4 \\\\ x &>-1 \end{aligned}
Solving the second inequality for x, we get:
8x+31<238x<8x<1\begin{aligned} 8x+31&<23\\\\ 8x&<-8\\\\ x&<-1 \end{aligned}
Graphically, we get:
Strangely, this means that there are no solutions to the compound inequality because there's no value of x that's both greater than negative one and less than negative one.
Want to learn more about compound inequalities that use AND statements? Check out this video.

Practice

Problem 1
Solve for x.
5, x, minus, 4, is greater than or equal to, 12, space, space, O, R, space, space, 12, x, plus, 5, is less than or equal to, minus, 4
Choose 1 answer:
Choose 1 answer:

Want more practice? Check out this exercise.