Question

solve the following inequality for g. write your answer in simplest form.

g - 5 ≤ 10g - 6

answer attempt 1 out of 2

g < ~
submit answer

Explanation:

Step1: Subtract \( g \) from both sides

To isolate the terms with \( g \), we subtract \( g \) from each side of the inequality \( g - 5 \leq 10g - 6 \). This gives us \( -5 \leq 9g - 6 \).

Step2: Add 6 to both sides

Next, we add 6 to both sides to get rid of the -6 on the right side. So, \( -5 + 6 \leq 9g - 6 + 6 \), which simplifies to \( 1 \leq 9g \).

Step3: Divide by 9

Now, we divide both sides by 9 to solve for \( g \). So, \( \frac{1}{9} \leq g \), or equivalently \( g \geq \frac{1}{9} \). Wait, let's check the steps again. Wait, in step 1, when we subtract \( g \) from both sides: \( g - 5 - g \leq 10g - 6 - g \) gives \( -5 \leq 9g - 6 \). Then add 6: \( -5 + 6 \leq 9g - 6 + 6 \) gives \( 1 \leq 9g \). Then divide by 9: \( \frac{1}{9} \leq g \), so \( g \geq \frac{1}{9} \). Wait, maybe I made a mistake in the inequality direction. Let's redo the original inequality: \( g - 5 \leq 10g - 6 \). Let's subtract \( 10g \) from both sides: \( g - 5 - 10g \leq 10g - 6 - 10g \), which is \( -9g - 5 \leq -6 \). Then add 5 to both sides: \( -9g - 5 + 5 \leq -6 + 5 \), so \( -9g \leq -1 \). Now, when we divide both sides by a negative number, the inequality sign flips. So, dividing both sides by -9: \( g \geq \frac{-1}{-9} \), which is \( g \geq \frac{1}{9} \). Wait, that's correct. Let's verify with the first method. Wait, in the first method, when we subtracted \( g \) from both sides, we got \( -5 \leq 9g - 6 \), then add 6: \( 1 \leq 9g \), then divide by 9: \( \frac{1}{9} \leq g \), which is the same as \( g \geq \frac{1}{9} \). So the correct solution is \( g \geq \frac{1}{9} \).

Answer:

\( g \geq \frac{1}{9} \)