Question

Which of the following values for x makes the inequality 4x+1−−−−−√ ≤5.83
true?(1 point)
Responses

8
8

9
9

10
10

9.5

Answer 1

To determine which value of \( x \) satisfies the inequality \( \sqrt{4x + 1} \leq 5.83 \), we can follow these steps:

1. **Square both sides** to eliminate the square root (note that squaring maintains the inequality because both sides are non-negative):
\[
4x + 1 \leq (5.83)^2
\]

2. **Calculate \( (5.83)^2 \)**:
\[
(5.83)^2 = 33.9889
\]

3. Substitute back into the inequality:
\[
4x + 1 \leq 33.9889
\]

4. **Subtract 1 from both sides**:
\[
4x \leq 33.9889 - 1
\]
\[
4x \leq 32.9889
\]

5. **Divide both sides by 4**:
\[
x \leq \frac{32.9889}{4}
\]
\[
x \leq 8.247225
\]

Now we can check each option:

1. **For \( x = 8 \)**:
\[
\sqrt{4(8) + 1} = \sqrt{32 + 1} = \sqrt{33} \approx 5.744 < 5.83 \quad \text{(True)}
\]

2. **For \( x = 9 \)**:
\[
\sqrt{4(9) + 1} = \sqrt{36 + 1} = \sqrt{37} \approx 6.083 > 5.83 \quad \text{(False)}
\]

3. **For \( x = 10 \)**:
\[
\sqrt{4(10) + 1} = \sqrt{40 + 1} = \sqrt{41} \approx 6.403 > 5.83 \quad \text{(False)}
\]

4. **For \( x = 9.5 \)**:
\[
\sqrt{4(9.5) + 1} = \sqrt{38 + 1} = \sqrt{39} \approx 6.244 > 5.83 \quad \text{(False)}
\]

The only value that satisfies the inequality \( \sqrt{4x + 1} \leq 5.83 \) is \( x = 8 \). Thus, the correct answer is:

**8**.