Question

27

Which values are solutions to th
compound inequality below? Select all the
correct answers.
-1 ≤3-x < 4
A. -2
B. -1
C. 0
D. 1
E. 2
F. 3
G. 4
H. 5

Answer 1

`-1 < x \leq 4`

C, D, E, F, G

Explanation:

Given compound inequality:

`-1 \leq 3-x < 4`

If a ≤ u < b then a ≤ u and u < b:

`\implies -1 \leq 3-x \;\; \textsf{and} \;\; 3-x < 4`

Solve each inequality:

`\begin{aligned}\underline{\sf Case \; 1}\\-1& \leq 3-x\\x-1 & \leq 3\\x&\leq 4\end{aligned}`
`\begin{aligned}\underline{\sf Case \; 2}\\3-x& < 4\\-x & < 1\\x& > -1\end{aligned}`

Combine the intervals:

`-1 < x \leq 4`

Therefore, the values that are solutions to the compound inequality from the given answer options are:

• C. x = 0
• D. x = 1
• E. x = 2
• F. x = 3
• G. x = 4

Answer 2

• C) 0, D) 1, E) 2, F) 3, G) 4

-----------------------------------

Given inequality:

• - 1 ≤ 3 - x < 4

Simplify it as below.

Add -3 to all sides:

• - 1 - 3 ≤ - x < 4 - 3
• -4 ≤ - x < 1

Multiply all sides by - 1, it flips the inequality signs:

• 4 ≥ x > - 1

Show in reverse order:

• - 1 < x ≤ 4 or
• x ∈ (- 1, 4]

Now verify the options.

Correct choices are C, D, E, F, G.