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Compound inequalities review

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 #7854ab, start text, space, O, R, space, end text, end color #7854ab, space, x, is greater than, 5
A number line from negative two to eight by ones. There is an open circle at three with an arrow to the left. There is an open circle at five with an arrow to the right.
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 #e07d10, start text, space, A, N, D, space, end text, end color #e07d10, 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:
A number line from negative two to eight by ones. There is an open circle at zero with an arrow to the right. There is an open circle at four with an arrow to the left.
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 #7854ab, start text, space, O, R, space, end text, end color #7854ab, 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:
A number line from negative one to five by ones. There is an open circle at one with an arrow to the right. There is a closed circle at two with an arrow to the right.
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 #e07d10, start text, space, A, N, D, end text, end color #e07d10, 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:
A number line from negative three to three by one. There is an open circle at negative one with an arrow to the left and to the right.
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, start text, space, O, R, space, end text, 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.

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