Question
3. Solve for \( x \) : \[ \frac{x}{5}-1 \leq 2-\frac{x}{10} \]
Ask by Guerrero George. in South Africa
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( x \leq 10 \)
Solution
1. Multiply both sides of the inequality by \(10\) to eliminate the fractions:
\[
10\left(\frac{x}{5}-1\right) \leq 10\left(2-\frac{x}{10}\right)
\]
2. Simplify each term:
\[
2x - 10 \leq 20 - x
\]
3. Add \(x\) to both sides to combine the \(x\) terms:
\[
3x - 10 \leq 20
\]
4. Add \(10\) to both sides:
\[
3x \leq 30
\]
5. Divide both sides by \(3\):
\[
x \leq 10
\]
The solution is \( x \leq 10 \).
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Simplify this solution Beyond the Answer
To solve the inequality \( \frac{x}{5}-1 \leq 2-\frac{x}{10} \), first, let's eliminate the fractions by finding a common denominator, which is 10. Multiply every term by 10: \[ 10 \left(\frac{x}{5}\right) - 10 \cdot 1 \leq 10 \cdot 2 - 10 \left(\frac{x}{10}\right) \] This simplifies to: \[ 2x - 10 \leq 20 - x \] Now, add \( x \) to both sides: \[ 2x + x - 10 \leq 20 \] This results in: \[ 3x - 10 \leq 20 \] Next, add 10 to both sides: \[ 3x \leq 30 \] Finally, divide by 3: \[ x \leq 10 \] Thus, the solution to the inequality is: \[ x \leq 10 \]
