Let's solve the problem step by step.
### Step 1: Identify the slope and [tex]\(y\)[/tex]-intercept.
The inequality given is [tex]\(y > 2x - 5\)[/tex].
The equation of a line in slope-intercept form is [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept.
From [tex]\(y > 2x - 5\)[/tex]:
- The slope [tex]\(m = 2\)[/tex]
- The [tex]\(y\)[/tex]-intercept [tex]\(b = -5\)[/tex]
### Step 2: Fill in the table with some values of [tex]\(x\)[/tex] and the corresponding boundary line values of [tex]\(y\)[/tex].
Next, we’ll determine the values of [tex]\(y\)[/tex] for different values of [tex]\(x\)[/tex], using the equation [tex]\(y = 2x - 5\)[/tex]:
[tex]\[
\begin{array}{|l|l|}
\hline
x & y \\
\hline
-3 & -11 \\
-2 & -9 \\
-1 & -7 \\
0 & -5 \\
1 & -3 \\
2 & -1 \\
3 & 1 \\
\hline
\end{array}
\][/tex]
This table lists the [tex]\(y\)[/tex] values from the line equation [tex]\(y = 2x - 5\)[/tex] for the chosen [tex]\(x\)[/tex] values.
### Step 3: Graph the boundary line and the solution set.
To graph the inequality [tex]\(y > 2x - 5\)[/tex]:
1. Plot the boundary line [tex]\(y = 2x - 5\)[/tex]:
- This is a straight line with a slope of 2 and a [tex]\(y\)[/tex]-intercept of [tex]\(-5\)[/tex]. Use the points from the table to plot this line.
2. Choose a test point (e.g., [tex]\((0,0)\)[/tex], which is not on the line):
- Substitute [tex]\(x = 0\)[/tex] and [tex]\(y = 0\)[/tex] into the inequality:
[tex]\[
0 > 2(0) - 5 \implies 0 > -5
\][/tex]
- The test point [tex]\((0,0)\)[/tex] satisfies the inequality, so this is part of the solution set.
3. Shade the region above the boundary line:
- Since the inequality is [tex]\(y > 2x - 5\)[/tex], shade the region above the boundary line (do not include the line itself, since the inequality is strict).
This is how you graph the solution set for the inequality [tex]\(y > 2x - 5\)[/tex].
