To convert the given system of inequalities into slope-intercept form, we need to re-arrange each inequality such that it follows the format [tex]\( y \leq mx + b \)[/tex] or [tex]\( y \geq mx + b \)[/tex]. Let's go through each inequality step-by-step:
1. Convert [tex]\( 4x - 5y \leq 1 \)[/tex] into slope-intercept form:
- Start by isolating [tex]\( y \)[/tex]. Subtract [tex]\( 4x \)[/tex] from both sides:
[tex]\[
-5y \leq -4x + 1
\][/tex]
- Next, divide every term by [tex]\( -5 \)[/tex]. Remember, dividing or multiplying both sides of an inequality by a negative number reverses the direction of the inequality:
[tex]\[
y \geq \frac{4}{5}x - \frac{1}{5}
\][/tex]
2. Convert [tex]\( \frac{1}{2}y - x \leq 3 \)[/tex] into slope-intercept form:
- First, eliminate the fraction by multiplying every term by 2:
[tex]\[
y - 2x \leq 6
\][/tex]
- Now, isolate [tex]\( y \)[/tex] by adding [tex]\( 2x \)[/tex] to both sides:
[tex]\[
y \leq 2x + 6
\][/tex]
The inequalities in slope-intercept form are:
[tex]\[
y \geq \frac{4}{5}x - \frac{1}{5}
\][/tex]
and
[tex]\[
y \leq 2x + 6
\][/tex]
So, the correct choice is:
[tex]\[
y \geq \frac{4}{5}x - \frac{1}{5}
\][/tex]
[tex]\[
y \leq 2x + 6
\][/tex]
