Question

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

9.5

8

10

9

Answer 1

We want to solve the inequality:

\[
\sqrt{4x + 1} \leq 5.83
\]

First, square both sides to eliminate the square root (keeping in mind that both sides are non-negative since we're dealing with a square root):

\[
4x + 1 \leq (5.83)^2
\]

Calculating \( (5.83)^2 \):

\[
(5.83)^2 = 33.9889
\]

So the inequality becomes:

\[
4x + 1 \leq 33.9889
\]

Next, subtract 1 from both sides:

\[
4x \leq 33.9889 - 1
\]
\[
4x \leq 32.9889
\]

Now divide both sides by 4:

\[
x \leq \frac{32.9889}{4}
\]
\[
x \leq 8.247225
\]

Now we can check the provided values: 9.5, 8, 10, and 9 to see which satisfies the inequality \( x \leq 8.247225 \).

- For \( x = 9.5 \): \( 9.5 \not\leq 8.247225 \) (false)
- For \( x = 8 \): \( 8 \leq 8.247225 \) (true)
- For \( x = 10 \): \( 10 \not\leq 8.247225 \) (false)
- For \( x = 9 \): \( 9 \not\leq 8.247225 \) (false)

The only value that makes the inequality \( \sqrt{4x + 1} \leq 5.83 \) true is:

\[
\boxed{8}
\]