To convert the given system of inequalities to slope-intercept form ([tex]\(y = mx + b\)[/tex]), let's go through each inequality step-by-step:
1. First Inequality: [tex]\(4x - 5y \leq 1\)[/tex]
We need to solve this for [tex]\(y\)[/tex].
[tex]\[
4x - 5y \leq 1
\][/tex]
Subtract [tex]\(4x\)[/tex] from both sides:
[tex]\[
-5y \leq -4x + 1
\][/tex]
Divide by [tex]\(-5\)[/tex], and remember to reverse the inequality since we are dividing by a negative number:
[tex]\[
y \geq \frac{4}{5}x - \frac{1}{5}
\][/tex]
Thus, the first inequality in slope-intercept form is:
[tex]\[
y \geq \frac{4}{5}x - \frac{1}{5}
\][/tex]
2. Second Inequality: [tex]\(\frac{1}{2}y - x \leq 3\)[/tex]
Again, solve this for [tex]\(y\)[/tex].
[tex]\[
\frac{1}{2}y - x \leq 3
\][/tex]
Add [tex]\(x\)[/tex] to both sides:
[tex]\[
\frac{1}{2}y \leq x + 3
\][/tex]
Multiply by [tex]\(2\)[/tex] to clear the fraction:
[tex]\[
y \leq 2x + 6
\][/tex]
Thus, the second inequality in slope-intercept form is:
[tex]\[
y \leq 2x + 6
\][/tex]
Now, compare these derived inequalities with the available choices:
- [tex]\(y \leq \frac{4}{5} x - \frac{1}{5}\)[/tex]
- [tex]\(y \leq 2 x + 6\)[/tex]
- [tex]\(y \geq \frac{4}{5} x - \frac{1}{5}\)[/tex]
- [tex]\(y \leq 2 x+ 6\)[/tex]
- [tex]\(y \geq-\frac{4}{5} x+\frac{1}{5}\)[/tex]
- [tex]\(y \geq 2 x + 6\)[/tex]
The inequalities from the choices that match our derived inequalities are:
- [tex]\(y \geq \frac{4}{5} x - \frac{1}{5}\)[/tex]
- [tex]\(y \leq 2 x + 6\)[/tex]
Therefore, the given inequalities in slope-intercept form are:
[tex]\[ y \geq \frac{4}{5} x - \frac{1}{5} \][/tex]
[tex]\[ y \leq 2 x + 6 \][/tex]
