Question

solve $54 - 10x \leq 20 + 7x$. \
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\bigcirc a. $x \leq -2$ \
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\bigcirc b. $x \geq -2$ \
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\bigcirc c. $x \geq 2$ \
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\bigcirc d. $x \leq 2$

Explanation:

Step1: Add \(10x\) to both sides

To isolate the variable terms, we add \(10x\) to each side of the inequality \(54 - 10x\leq20 + 7x\). This gives us \(54\leq20 + 17x\).

Step2: Subtract 20 from both sides

Next, we subtract 20 from both sides to get \(34\leq17x\).

Step3: Divide both sides by 17

Dividing both sides of the inequality \(34\leq17x\) by 17 (since 17 is positive, the inequality sign remains the same), we obtain \(2\leq x\), which is equivalent to \(x\geq2\). Wait, no, wait, let's check again. Wait, original inequality: \(54 - 10x\leq20 + 7x\). Let's do it again.

Wait, step 1: Add \(10x\) to both sides: \(54\leq20 + 17x\). Then step 2: Subtract 20: \(54 - 20\leq17x\) → \(34\leq17x\). Then step 3: Divide by 17: \(34/17\leq x\) → \(2\leq x\) → \(x\geq2\). Wait, but let's check the options. Option C is \(x\geq2\). Wait, but let's verify with a value. Let's take \(x = 2\): left side \(54 - 20 = 34\), right side \(20 + 14 = 34\), so \(34\leq34\) which is true. Take \(x = 3\): left \(54 - 30 = 24\), right \(20 + 21 = 41\), \(24\leq41\) true. Take \(x = 1\): left \(54 - 10 = 44\), right \(20 + 7 = 27\), \(44\leq27\) false. So \(x\geq2\) is correct. Wait, but let's check the steps again. Wait, maybe I made a mistake earlier. Wait, original inequality: \(54 - 10x\leq20 + 7x\). Let's move all x terms to right and constants to left. So add \(10x\) to both sides: \(54\leq20 + 17x\). Subtract 20: \(34\leq17x\). Divide by 17: \(2\leq x\) → \(x\geq2\). Yes, that's correct. So the answer should be C. \(x\geq2\).

Wait, but let's check the options again. The options are A. \(x\leq - 2\), B. \(x\geq - 2\), C. \(x\geq2\), D. \(x\leq2\). So the correct answer is C. \(x\geq2\).

Wait, but let's do the inequality again carefully.

Given \(54 - 10x\leq20 + 7x\)

Add \(10x\) to both sides:

\(54 - 10x + 10x\leq20 + 7x + 10x\)

\(54\leq20 + 17x\)

Subtract 20 from both sides:

\(54 - 20\leq20 + 17x - 20\)

\(34\leq17x\)

Divide both sides by 17 (since 17 > 0, inequality direction remains):

\(\frac{34}{17}\leq\frac{17x}{17}\)

\(2\leq x\)

Which is \(x\geq2\). So the correct option is C. \(x\geq2\).

Answer:

C. \(x\geq2\)