Question

How can you solve a compound inequality and graph the solution set?

Answer 1

Answer:

Here's how I would do it.  

Step-by-step explanation:

A compound inequality contains two or more inequalities that are separated by "and" or "or".

Examples are "3x - 1 <8 or x - 5 ≥ 0" and "-10 < 2 + x  ≤ -1". The second example is an "and" inequality.

Let's solve the second example.

1. Solve each inequality separately

$\begin{array}{rcl}-10 & < & 2 + x\\-12 & < & x\\\end{array}$

$\begin{array}{rcl}2+x & \leq & -1\\x& \leq &-3\\\end{array}$

You can write the answer as "-12 < x and x ≤ -3" or "-12 < x ≤ -3" or (-12, -3]."

2. Check your solution.

Pick a point between -3 and -12 and see if it satisfies both inequalities, say, -10

-10 < 2 -10     2 + (-10) ≤ -1

-10 < -8              2 - 10 ≤ -1

                             -10 ≤ -1

The point satisfies both inequalities.

3. Graph the compound inequality

You can plot the graphs on a number line.

The graph of the first inequality is represented by the blue line in the first diagram below.

It starts at -3 and extends to the left. The solid circle indicates that -3 is part of the solution set.

The graph of the second inequality is represented by the red line in the second diagram.

It starts at -12 and extends to the right. The open circle indicates that -12 is not part of the solution set.

The solution to the compound inequality is the region in which the two graphs overlap — that is, from -3 down to but not including -12.

The graph is shown in the third diagram.