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

Question

true?(1 point)
Responses

8
8

9
9

10
10

9.5

Answer 1

To determine which value of $x$ satisfies the inequality $\sqrt{4x+1}\le 5.83$, 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): $$4x+1\le (5.83{)}^2$$

2. Calculate $(5.83{)}^2$: $$(5.83{)}^2=33.9889$$

3. Substitute back into the inequality: $$4x+1\le 33.9889$$

4. Subtract 1 from both sides: $$4x\le 33.9889-1$$ $$4x\le 32.9889$$

5. Divide both sides by 4: $$x\le \frac{32.9889}{4}$$ $$x\le 8.247225$$

Now we can check each option:

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

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

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

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

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

8.