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Study Guides > MATH 1314: College Algebra

Understanding Compound Inequalities

A compound inequality includes two inequalities in one statement. A statement such as 4<x64<x\le 6 means 4<x4<x and x6x\le 6. There are two ways to solve compound inequalities: separating them into two separate inequalities or leaving the compound inequality intact and performing operations on all three parts at the same time. We will illustrate both methods.

Example 7: Solving a Compound Inequality

Solve the compound inequality: 32x+2<63\le 2x+2<6.

Solution

The first method is to write two separate inequalities: 32x+23\le 2x+2 and 2x+2<62x+2<6. We solve them independently.
32x+2and2x+2<612x2x<412xx<2\begin{array}{lll}3\le 2x+2\hfill & \text{and}\hfill & 2x+2<6\hfill \\ 1\le 2x\hfill & \hfill & 2x<4\hfill \\ \frac{1}{2}\le x\hfill & \hfill & x<2\hfill \end{array}
Then, we can rewrite the solution as a compound inequality, the same way the problem began.
12x<2\frac{1}{2}\le x<2
In interval notation, the solution is written as [12,2)\left[\frac{1}{2},2\right). The second method is to leave the compound inequality intact, and perform solving procedures on the three parts at the same time.
32x+2<612x<4Isolate the variable term, and subtract 2 from all three parts.12x<2Divide through all three parts by 2.\begin{array}{ll}3\le 2x+2<6\hfill & \hfill \\ 1\le 2x<4\hfill & \text{Isolate the variable term, and subtract 2 from all three parts}.\hfill \\ \frac{1}{2}\le x<2\hfill & \text{Divide through all three parts by 2}.\hfill \end{array}
We get the same solution: [12,2)\left[\frac{1}{2},2\right).

Try It 7

Solve the compound inequality 4<2x8104<2x - 8\le 10. Solution

Example 8: Solving a Compound Inequality with the Variable in All Three Parts

Solve the compound inequality with variables in all three parts: 3+x>7x2>5x103+x>7x - 2>5x - 10.

Solution

Lets try the first method. Write two inequalities:
3+x>7x2and7x2>5x103>6x22x2>105>6x2x>856>xx>4x<564<x\begin{array}{lll}3+x> 7x - 2\hfill & \text{and}\hfill & 7x - 2> 5x - 10\hfill \\ 3> 6x - 2\hfill & \hfill & 2x - 2> -10\hfill \\ 5> 6x\hfill & \hfill & 2x> -8\hfill \\ \frac{5}{6}> x\hfill & \hfill & x> -4\hfill \\ x< \frac{5}{6}\hfill & \hfill & -4< x\hfill \end{array}
The solution set is 4<x<56-4<x<\frac{5}{6} or in interval notation (4,56)\left(-4,\frac{5}{6}\right). Notice that when we write the solution in interval notation, the smaller number comes first. We read intervals from left to right, as they appear on a number line.
A number line with the points -4 and 5/6 labeled. Dots appear at these points and a line connects these two dots. Figure 3

Try It 8

Solve the compound inequality: 3y<45y<5+3y3y<4 - 5y<5+3y. Solution

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