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|x-10|>|x-30|
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12 May 2025, 09:11
12 May 2025, 09:11
Expert Reply
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Question Stats:
42% (01:14) correct
57% (00:50) wrong based on 7 sessions
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\(|x-10|>|x-30|\)
Quantity A |
Quantity B |
x |
20 |
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Re: |x-10|>|x-30|
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14 May 2025, 00:40
14 May 2025, 00:40
1
to these kindof inequality, do squaring both sides, since both sides are non-negative
squaring both side: (x−10)^2 > (x−30) ^2
x^2 −20x + 100 > x^2 − 60x + 900
Subtracting x^2 from both sides:
−20x+100>−60x+900
Adding 60x to both sides: 40x+100>900
Simply to get: x>20
So, x must be greater than 20
Hence, A is bigger
squaring both side: (x−10)^2 > (x−30) ^2
x^2 −20x + 100 > x^2 − 60x + 900
Subtracting x^2 from both sides:
−20x+100>−60x+900
Adding 60x to both sides: 40x+100>900
Simply to get: x>20
So, x must be greater than 20
Hence, A is bigger
Re: |x-10|>|x-30|
[#permalink]
15 May 2025, 04:15
15 May 2025, 04:15
Expert Reply
1. Square both sides to eliminate absolute values:
$$
\((x-10)^2>(x-30)^2\)
$$
2. Expand both sides:
$$
\(x^2-20 x+100>x^2-60 x+900\)
$$
3. Simplify:
$$
\(\begin{aligned}
-20 x+100 & >-60 x+900 \\
40 x & >800 \\
x & >20
\end{aligned}\)
$$
This confirms the solution $\(x>20\)$.
Step 3: Compare Quantity A ( $x$ ) and Quantity B (20)
From the solution to the inequality:
$$
\(x>20\)
$$
This means:
- Quantity A ( $x$ ) is greater than Quantity B (20) for all valid $x$.
Verification with Examples
1. Example 1: $\(x=25\)$
- Check inequality: $\(|25-10|=15>|25-30|=5 \rightarrow\)$ True
- Comparison: $\(25>20 \rightarrow A>B\)$
2. Example 2: $\(x=35\)$
- Check inequality: $\(|35-10|=25>|35-30|=5 \rightarrow\)$ True
- Comparison: $\(35>20 \rightarrow A>B\)$
3. Example 3: $\(x=20\)$
- Check inequality: $\(|20-10|=10 \ngtr|20-30|=10 \rightarrow\)$ False
- Not part of the solution set.
Conclusion
For all $x$ satisfying $\(|x-10|>|x-30|, x>20\)$. Therefore:
Quantity $\(\mathbf{A}(x)\)$ is greater than Quantity $\(\mathbf{B}(20)\)$.
$$
\((x-10)^2>(x-30)^2\)
$$
2. Expand both sides:
$$
\(x^2-20 x+100>x^2-60 x+900\)
$$
3. Simplify:
$$
\(\begin{aligned}
-20 x+100 & >-60 x+900 \\
40 x & >800 \\
x & >20
\end{aligned}\)
$$
This confirms the solution $\(x>20\)$.
Step 3: Compare Quantity A ( $x$ ) and Quantity B (20)
From the solution to the inequality:
$$
\(x>20\)
$$
This means:
- Quantity A ( $x$ ) is greater than Quantity B (20) for all valid $x$.
Verification with Examples
1. Example 1: $\(x=25\)$
- Check inequality: $\(|25-10|=15>|25-30|=5 \rightarrow\)$ True
- Comparison: $\(25>20 \rightarrow A>B\)$
2. Example 2: $\(x=35\)$
- Check inequality: $\(|35-10|=25>|35-30|=5 \rightarrow\)$ True
- Comparison: $\(35>20 \rightarrow A>B\)$
3. Example 3: $\(x=20\)$
- Check inequality: $\(|20-10|=10 \ngtr|20-30|=10 \rightarrow\)$ False
- Not part of the solution set.
Conclusion
For all $x$ satisfying $\(|x-10|>|x-30|, x>20\)$. Therefore:
Quantity $\(\mathbf{A}(x)\)$ is greater than Quantity $\(\mathbf{B}(20)\)$.
