Compound Inequalities
Inequalities tell you something about a variable: what that variable is greater than or less than.
Compound inequalities tell you two things about a variable: two numbers that the variable is greater than or less than.
Sometimes a compound in equality will tell you what two numbers a variable lies between. Other times, a compound in equality will give you two distinct areas of the number line that a variable could inhabit.
Compound inequalities that show that you a variable could lie in one of two distinct parts of the number line are called OR inequalities. They say that a number is greater than something OR less than something else.
Example of an OR inequality:
$x < -4 \text{ OR } x \geq 20$ This compound in equality graphs like this: $\dfrac{\leftarrow \!\! \circ \quad \bullet \!\! \rightarrow}{\quad \, -4 \quad 20\quad}$
Note two things about the OR inequality: The inequality symbols face in opposite directions. Also, when graphed, the rays of the graph point in opposite directions. OR inequalities are all about opposites.
When you have a multi-step compound OR inequality, you solve each inequality on its own, just like a non-compound inequality. Essentially, an OR compound inequality is two inequalities, that refer to the same variable. Solve each, but graph them on the same number line.
Compound inequalities that show that you a variable lies between two numbers are called AND inequalities. They say that a number is greater than a low number and less than a higher number, thus the variable's value lies somewhere between those two numbers.
$-4 < x \leq 20$ This compound in equality graphs like this: $\dfrac{\circ \!\!\rightarrow \!\! \leftarrow \!\! \bullet}{-4 \qquad 20}$
Note two things about the AND inequality: The inequality symbols face in the same direction, and no word is necessary: the two inequalities fit together in a single sentence, "x is greater than -4 and less than 20." Also, when graphed, the rays of the graph point toward each other. They can also be written as a single line segment between the two circles.
Solving multistep compound AND inequalities is a little different from (but follows the same rules as) solving OR inequalities.
When solving a compound AND inequality, isolate the variable in the middle. And, whatever you do to isolate the variable in the middle, you must do to BOTH sides of the inequality.
Let's try to solve a compound AND inequality:
$\eqalign{-3 < 2x+1 < 19\\-1 \quad -1 \quad -1\\-4 < 2x < 18\\\div 2 \, \div 2 \, \div 2\\-2 < x < 9}$
It's just like solving a regular equation or inequality except you have to keep three parts of the inequality in balance.
Practice Problems:
Compound Inequalities
Solve the inequalities:
- $\eqalign{13>x-1>5}$
- $\eqalign{-2<y+5<10}$
- $\eqalign{-15 \leq 2z-3 \leq 6}$
- $\eqalign{22>4x+10>18}$
- $\eqalign{-1<-3a+7<90}$
- $\eqalign{3-1 \ge 6b-1 \ge 1}$
- $\eqalign{40>-8c+2>6-10}$
- $\eqalign{140+6>12v+2>62}$
- $\eqalign{15<-x+10<54}$
- $\eqalign{32 \ge -14g-10 \ge 4}$
Answer Key: