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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

1 answer

To solve the inequality \( \sqrt{4x + 1} \leq 5.83 \), we can follow these steps:

  1. Square both sides of the inequality (since both sides are positive, this preserves the inequality): \[ 4x + 1 \leq (5.83)^2 \]

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

Now, we evaluate the given options to see which values satisfy \( x \leq 8.247225 \):

  • 8: This value is less than 8.247225 and satisfies the inequality.
  • 9: This value is greater than 8.247225 and does not satisfy the inequality.
  • 10: This value is greater than 8.247225 and does not satisfy the inequality.
  • 9.5: This value is greater than 8.247225 and does not satisfy the inequality.

Thus, the only value that makes the inequality \( \sqrt{4x + 1} \leq 5.83 \) true is 8.