Question

which equation is equivalent to \\(\sqrt{x} + 11 = 15\\)?\
options:\
\\(x + 121 = 225\\)\
\\(x + 11 = 225\\)\
\\(\sqrt{x} = 15 - 11\\)\
\\(\sqrt{x} = 15 + 11\\)

Explanation:

Step1: Isolate the square root term

Start with the equation \(\sqrt{x} + 11 = 15\). Subtract 11 from both sides to isolate \(\sqrt{x}\).

$$ \sqrt{x} + 11 - 11 = 15 - 11 $$

Simplifying both sides gives \(\sqrt{x} = 15 - 11\). Wait, no, let's check the options again. Wait, maybe I made a mistake. Wait, the correct way: Let's start over. The original equation is \(\sqrt{x}+11 = 15\). To isolate \(\sqrt{x}\), subtract 11: \(\sqrt{x}=15 - 11\)? No, wait, no—wait, the options: Wait, maybe the correct step is to square both sides? Wait, no, the options given: Let's list the options. The options are:

1. \(x + 121 = 225\)
2. \(x + 11 = 225\)
3. \(\sqrt{x}=15 - 11\)
4. \(\sqrt{x}=15 + 11\)

Wait, let's do the correct manipulation. Starting with \(\sqrt{x}+11 = 15\). Subtract 11 from both sides: \(\sqrt{x}=15 - 11\) (which is option 3). But wait, maybe the intended step is to square both sides? Wait, no, let's check the options again. Wait, maybe I misread the original equation. Wait, the original equation is \(\sqrt{x}+11 = 15\). Let's solve for \(\sqrt{x}\): \(\sqrt{x}=15 - 11\), which is \(\sqrt{x}=4\)? No, 15-11 is 4? Wait, no, 15-11 is 4? Wait, 15-11=4. But maybe the options are different. Wait, the options: Let's check the image again. The options are:

• \(x + 121 = 225\)
• \(x + 11 = 225\)
• \(\sqrt{x}=15 - 11\)
• \(\sqrt{x}=15 + 11\)

Wait, let's do the correct algebraic manipulation. Starting with \(\sqrt{x}+11 = 15\). Subtract 11 from both sides: \(\sqrt{x}=15 - 11\) (so that's option 3: \(\sqrt{x}=15 - 11\)). But wait, maybe the question is to find the equivalent equation by squaring? Wait, no, the options don't have a squared version? Wait, no, let's check again. Wait, maybe I made a mistake. Wait, the original equation is \(\sqrt{x}+11 = 15\). If we square both sides, we get \((\sqrt{x}+11)^2 = 15^2\), which expands to \(x + 22\sqrt{x}+121 = 225\), which is not one of the options. So the correct equivalent equation by subtracting 11 is \(\sqrt{x}=15 - 11\), which is the third option (the one with \(\sqrt{x}=15 - 11\)). Wait, but let's check the options again. The third option (the second box in the left column) is \(\sqrt{x}=15 - 11\). So that's the equivalent equation.

Wait, but let's confirm. The original equation is \(\sqrt{x}+11 = 15\). Subtract 11 from both sides: \(\sqrt{x}=15 - 11\). So that's the equivalent equation. So the correct option is \(\sqrt{x}=15 - 11\).

Step2: Verify the manipulation

Subtracting 11 from both sides of an equation preserves equality. So starting with \(\sqrt{x}+11 = 15\), subtracting 11 from the left side removes the +11, and subtracting 11 from the right side gives 15 - 11. Thus, \(\sqrt{x}=15 - 11\) is equivalent.

Answer:

\(\sqrt{x} = 15 - 11\) (the option with \(\sqrt{x}=15 - 11\))