Solve Simple and Compound Linear Inequalities

Example 7

Solving a Compound Inequality

Solve the compound inequality: $3 \leq 2 x+2 < 6$.

Solution

The first method is to write two separate inequalities: $3 \leq 2 x+2$ and $2 x+2 < 6$. We solve them independently.

$3 \leq 2 x+2 \quad$ and $\quad 2 x+2 < 6$

$1 \leq 2 x \quad 2 x < 4$

$\frac{1}{2} \leq x \quad x < 2$

Then, we can rewrite the solution as a compound inequality, the same way the problem began.

$\frac{1}{2} \leq x < 2$

In interval notation, the solution is written as $\left[\frac{1}{2}, 2\right)$.

The second method is to leave the compound inequality intact, and perform solving procedures on the three parts at the same time.

$\begin{align} 3 \leq 2 x+2 < 6 \\ 1 \leq 2 x < 4 & \qquad \qquad \text{Isolate the variable term, and subtract 2 from all three parts.} \\ \frac{1}{2} \leq x < 2 & \qquad \qquad \text{Divide through all three parts by 2}.\end{align}$

We get the same solution: $\left[\frac{1}{2}, 2\right)$.

Try It #7

Solve the compound inequality: $4 < 2 x-8 \leq 10$.

Example 8

Solving a Compound Inequality with the Variable in All Three Parts

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

Solution

Let's try the first method. Write two inequalities:

 $\begin{array}{ccr} 3+x > 7 x-2 & \qquad \quad \text { and }& \qquad 7 x-2 > 5 x-10\\ 3 > 6 x-2 & & 2 x-2 > -10\\ 5 > 6 x & & 2 x > -8\\ \frac{5}{6} > x & & x > -4\\ x < \frac{5}{6} & & -4 < x \end{array}$

The solution set is $-4 < x < \frac{5}{6}$ or in interval notation $\left(-4, \frac{5}{6}\right)$. 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. See Figure 3.

Figure 3

Try It #8

Solve the compound inequality: $3y < 4-5y < 5+3y$.