d Create an or compound inequality because the absolute value is greater than or equal to the given value.
A
aInequality: x ≤ 12
Number Line:
B
bInequality: - 10 < x < 10
Number Line:
C
cInequality: x<0
Number Line:
D
dInequality: x< - 5 or x>1
Number Line:
Practice makes perfect
aInequalities can be solved in the same way as equations, by performing inverse operations on both sides until the variable is isolated. The only difference is that when you divide or multiply by a negative number, you must reverse the inequality sign.
This inequality tells us that all values less than or equal to 12 will satisfy the inequality.
Below we demonstrate the inequality by graphing the solution set on a number line. Notice that x can equal 12, which we show with a closed circle on the number line.
b We are asked to solve the given inequality and graph the solution set on the number line.
|x|-3 < 7
Let's begin by gathering all constant terms on the right-hand side of the inequality.
|x|-3 < 7 ⇔ |x|<10
Now we can create a compound inequality by removing the absolute value. In this case the solution set is any number less than 10 away from the midpoint in the positive direction and any number less than 10 away from the midpoint in the negative direction.
Absolute Value Inequality:& |x| < 10
Compound Inequality:& - 10< x < 10
The graph of this inequality includes all values from - 10 to 10, not inclusive. We show this by using open circles on the endpoints.
c Inequalities can be solved in the same way as equations, by performing inverse operations on both sides until the variable is isolated. The only difference is that when you divide or multiply by a negative number, you must reverse the inequality sign.
This inequality tells us that all values less than 0 will satisfy the inequality.
Below we demonstrate the inequality by graphing the solution set on a number line. Notice that x cannot equal 0, which we show with an open circle on the number line.
d We are asked to solve the given inequality and graph the solution set on the number line.
|x+2|> 3
To do this we will create a compound inequality by removing the absolute value. In this case, and since x+2 can be written as x-(- 2), the solution set contains the numbers that make the distance between x and - 2 greater than 3 in the positive direction or in the negative direction.
x+2 > 3 or x+2 <- 3
Let's isolate x in both of these cases before graphing the solution set.
This inequality tells us that all values less than - 5 will satisfy the inequality.
Solution Set
The solution to this type of compound inequality is the combination of the solution sets.
First Solution Set:& x>1
Second Solution Set:& x < - 5
Combined Solution Set:& x < - 5 or x>1
Graph
The graph of this inequality includes all values less than - 5 or greater than 1. Notice that the inequality is strict. We show this by keeping the endpoints open.
