Solve the inequalities and compare their solutions.
≤
Practice makes perfect
For this exercise, we will solve both inequalities and then compare their solutions. Then, we will determine which symbol the first inequality must have to ensure that the solution set of the compound inequality only has one value.
First Inequality:& 4(x-6) 2(x-10)
Second Inequality:& 5(x+2) ≥ 2(x+8)
We will begin by solving the second inequality.
The solution to the second inequality is x ≥ 2. We will now solve the first inequality. Note that we are allowed to ignore what is inside the box as long as we do not multiply or divide by a negative number.
Solving the first inequality gives x 2. Recall that we have already calculated the solution to the second inequality.
x ≥ 2
This means the second inequality is true only when x takes values 2 or larger. We will use the table below to show the number of solutions the compound inequality would have based on the different symbols for the first inequality.
x 2
Number of solutions
x<2
None
x>2
Infinitely many
x≤ 2
One
x≥ 2
Infinitely many
If the solution to the first inequality is x ≤ 2, and the solution to the second one is x ≥ 2, the compound inequality has exactly one solution, x=2.
