Which of the following values of satisfies the compound inequality
?
A compound inequality makes two demands on the same expression at once. Graph each demand in Desmos and look for the overlap; only values in both shaded regions work. Passing one side of the inequality isn't enough. Check whether each boundary is included before choosing a value.
Hints
- Hint 1
A compound inequality says the middle expression must satisfy both comparisons at once. Read the chain as one condition involving the left bound and another involving the right bound. What are those two conditions?
- Hint 2
An inequality graph shades the values that satisfy its condition. The overlap of two shaded regions satisfies both conditions. Where do the candidate values fall relative to that overlap?
Step-by-step
Approach 1: Graph both conditions in Desmos
Step 1Graph the lower condition
The left side of the compound inequality requires to be greater than . Type in Desmos. It shades ; the edge at is dashed because doesn't satisfy the strict inequality.
- Step 2
Find where both conditions hold
The right side also requires to be at most . Add . Desmos shades , including the solid edge at . The two shadings overlap where .
- Step 3
Check the candidate values
Plot , , , and . Each point's first coordinate is a candidate value of ; the places it on the horizontal axis. Only lies in the overlap, so satisfies both conditions. Choice B.
Approach 2: Solve the chain by hand
Step 1Remove the added 6
Subtract from all three parts of the chain. Subtraction keeps both inequality signs pointed the same way:
- Step 2
Get x alone
Divide all three parts by . Because is positive, neither sign flips:
- Step 3
Choose a value in the range
Check the choices against both ends of the range: . So satisfies the compound inequality. Choice B.