How can we use compound inequalities?
As previously mentioned, compound inequalities have at least two inequalities separated by an “and” or an “or”. If you are graphing a compound inequality that uses the “and”, then we must only include the intersection of the two inequalities. We usually see this as a range between two values, but we could also see it extending in one direction. You can see this compound inequality written in two ways:
x > 5 and x < 7
5 < x < 7
To solve a compound inequality written like the first one, we would solve them separately and find the intersection of the solutions. To solve a compound inequality written like the second one, you will solve by doing the operations on all sides of the inequality as shown in the solved examples. When we graph these on a number line, we must only graph the intersection of the solutions. This can result in a no solution situation.
If you are graphing a compound inequality that has an “or”, the solution represents the union of the graphs. These can have graphs facing opposite directions or in the same direction. These are usually written like:
x < 3 or x > 6.
To solve a compound inequality with an or, you will solve each inequality separately. To graph them, you will graph the solutions that will satisfy one or the other inequality. This can result in solutions representing all real numbers.
Sample Math Problems
1. Graph the solution to the compound inequality: x > -3 and x < 5
Solution:
The solution to x > -3 is shown in the picture in purple. The graph to the inequality x < 5 is shown in green. You can see that the two inequalities intersect between -3 and 5 and do not include the numbers -3 and 5. Therefore we use an open circle at -3 and 5, then shade the area between. This is shown in the diagram as a red line. The red line is the only solution to the compound inequality x > -3 and x < 5.
2. Solve the compound inequality 12 < 3x + 15 < 21 and graph the solution.
Solution: When you solve the compound inequality, you will isolate the x as with a regular inequality, but making sure to perform all operations throughout the inequality.
-1 < x < 2
3. Graph the solution to the compound inequality x < -3 or x > 5
Solution: Since there is an “or” between, we will identify the solutions for each of these and graph the union. For x < -3, the solution includes all values less than -3, as identified in purple on the diagram. For x > 5, the solution includes all values greater than 5.
Since this an “or”, both pieces are part of the solution as shown in the number line below.
4. Solve and graph the solution to the compound inequality
a)
Solution: For this one, we will solve each one individually, then graph the solution.
After we have solved this, we can then graph the solutions for numbers less than or equal to -9 and greater than or equal to 1 as shown below.
b)
