Sure, let's take a step-by-step approach to solving the inequality [tex]\(-5x + 2y > -8\)[/tex].
### Step 1: Understand the Inequality
The given inequality is [tex]\(-5x + 2y > -8\)[/tex]. Our goal is to describe the set of [tex]\((x, y)\)[/tex] pairs that satisfy this inequality.
### Step 2: Rearrange the Inequality
We can rearrange the inequality to express [tex]\(y\)[/tex] in terms of [tex]\(x\)[/tex]. Here’s how:
- First, add [tex]\(5x\)[/tex] to both sides of the inequality:
[tex]\[
-5x + 2y + 5x > -8 + 5x
\][/tex]
Simplifying this, we get:
[tex]\[
2y > 5x - 8
\][/tex]
- Next, divide each term by 2 to solve for [tex]\(y\)[/tex]:
[tex]\[
\frac{2y}{2} > \frac{5x}{2} - \frac{8}{2}
\][/tex]
This simplifies to:
[tex]\[
y > \frac{5}{2}x - 4
\][/tex]
### Step 3: Interpret the Inequality
The inequality [tex]\(y > \frac{5}{2}x - 4\)[/tex] represents a region in the coordinate plane.
- The line [tex]\(y = \frac{5}{2}x - 4\)[/tex] is the boundary line.
- Since the inequality is [tex]\(y > \frac{5}{2}x - 4\)[/tex], it represents the region above this line.
### Step 4: Graph the Solution
To graph this inequality:
1. Draw the line [tex]\(y = \frac{5}{2}x - 4\)[/tex].
- This line has a slope of [tex]\(\frac{5}{2}\)[/tex] and a y-intercept of [tex]\(-4\)[/tex].
2. Since the inequality is strict ([tex]\(>\)[/tex]), the line itself is not included in the solution, so it is drawn as a dashed line.
3. Shade the region above this line, because [tex]\(y\)[/tex] must be greater than the expression [tex]\(\frac{5}{2}x - 4\)[/tex].
### Solution Summary
The inequality [tex]\(-5x + 2y > -8\)[/tex] describes the region of the coordinate plane above the line [tex]\(y = \frac{5}{2}x - 4\)[/tex]. This region consists of all points [tex]\((x, y)\)[/tex] for which [tex]\(y\)[/tex] is greater than [tex]\(\frac{5}{2}x - 4\)[/tex].
In conclusion, the inequality [tex]\(-5x + 2y > -8\)[/tex] represents the region above the line [tex]\(y = \frac{5}{2}x - 4\)[/tex].
