To graph the solution set for the inequality [tex]\( 5x - 4y > 20 \)[/tex], follow these steps:
1. Rewrite the inequality in slope-intercept form (if needed):
The inequality [tex]\(5x - 4y > 20\)[/tex] can be rewritten as:
[tex]\[
-4y > -5x + 20
\][/tex]
To isolate [tex]\(y\)[/tex], divide each term by [tex]\(-4\)[/tex] (note that dividing by a negative number reverses the inequality sign):
[tex]\[
y < \frac{5}{4}x - 5
\][/tex]
2. Graph the boundary line:
The boundary line is [tex]\( y = \frac{5}{4}x - 5 \)[/tex]. Since the original inequality is [tex]\(5x - 4y > 20\)[/tex] and not [tex]\( \geq \)[/tex], the boundary line should be dashed.
- Find the intercepts:
- Y-intercept: Set [tex]\(x = 0\)[/tex]
[tex]\[
y = \frac{5}{4}(0) - 5 = -5
\][/tex]
So, the y-intercept is [tex]\((0, -5)\)[/tex].
- X-intercept: Set [tex]\(y = 0\)[/tex]
[tex]\[
0 = \frac{5}{4}x - 5 \implies \frac{5}{4}x = 5 \implies x = 4
\][/tex]
So, the x-intercept is [tex]\((4, 0)\)[/tex].
Plot these points on the graph, [tex]\((0, -5)\)[/tex] and [tex]\((4, 0)\)[/tex], and draw the dashed line that passes through them.
3. Determine the shading region:
Since the inequality is [tex]\( y < \frac{5}{4}x - 5 \)[/tex], you shade the area below the dashed line:
- Pick a test point that is not on the dashed line to determine the correct side for shading. A common test point is [tex]\((0, 0)\)[/tex].
- Substitute [tex]\((0, 0)\)[/tex] into the inequality [tex]\(y < \frac{5}{4}x - 5\)[/tex]:
[tex]\[
0 < \frac{5}{4}(0) - 5 \implies 0 < -5 \quad(\text{false})
\][/tex]
Since (0, 0) does not satisfy the inequality, you should shade the opposite side (below the line).
Using the provided Drawing Tools:
1. Select the Dashed Line tool and draw a dashed line through the points [tex]\((0, -5)\)[/tex] and [tex]\((4, 0)\)[/tex].
2. Select the Shaded Region tool and shade the area below the dashed line to represent [tex]\( y < \frac{5}{4}x - 5 \)[/tex].
That's how you form the correct answer on the provided graph!
