To graph the solution set of the first linear inequality [tex]\(7x + 5y < -35\)[/tex], let's follow these steps:
### Step 1: Convert the Inequality into an Equation
First, we need to deal with the related equation:
[tex]\[ 7x + 5y = -35 \][/tex]
### Step 2: Plot the Boundary Line
We will graph this equation as a line to divide the plane into two regions.
#### 2.1: Find the Intercepts
To make it easier to draw the line, we can find the x-intercept and y-intercept.
- X-intercept: Set [tex]\( y = 0 \)[/tex] in the equation [tex]\( 7x + 5y = -35 \)[/tex]:
[tex]\[
7x + 5(0) = -35 \implies 7x = -35 \implies x = -5
\][/tex]
So, the x-intercept is [tex]\( (-5, 0) \)[/tex].
- Y-intercept: Set [tex]\( x = 0 \)[/tex] in the equation [tex]\( 7x + 5y = -35 \)[/tex]:
[tex]\[
7(0) + 5y = -35 \implies 5y = -35 \implies y = -7
\][/tex]
So, the y-intercept is [tex]\( (0, -7) \)[/tex].
#### 2.2: Draw the Line
Using the intercepts [tex]\((-5, 0)\)[/tex] and [tex]\( (0, -7) \)[/tex], plot these points on the coordinate plane and draw a straight line through them. Ensure the line is dashed because the inequality is a strict inequality ([tex]\(<\)[/tex]), which means points on the line are not included in the solution.
### Step 3: Shade the Appropriate Region
Since the inequality is [tex]\( 7x + 5y < -35 \)[/tex], we need to determine which side of the line to shade.
#### 3.1: Test a Point Not on the Line
A common test point is the origin [tex]\((0, 0)\)[/tex], unless it lies on the line. Substitute [tex]\((0, 0)\)[/tex] into the inequality:
[tex]\[
7(0) + 5(0) < -35 \implies 0 < -35
\][/tex]
This statement is false, meaning the origin is not in the solution region. Therefore, the region that satisfies the inequality [tex]\( 7x + 5y < -35 \)[/tex] is the one that does not include the origin. Shade the region that is below the dashed line.
### Summary for Step 1
The graph of the first inequality [tex]\(7x + 5y < -35\)[/tex] consists of a dashed line passing through the points [tex]\((-5, 0)\)[/tex] and [tex]\( (0, -7) \)[/tex]. The region below this line (away from the origin) is shaded to represent all points [tex]\((x, y)\)[/tex] that satisfy this inequality.
Next, we will address the second inequality [tex]\( x \ge -2 \)[/tex].
