Answer :
To graph the linear inequality [tex]\( x - 2y \geq -12 \)[/tex], follow these step-by-step instructions to ensure a clear and accurate representation of the solution:
### Step 1: Rewrite the Inequality in Slope-Intercept Form
First, rewrite the inequality in the form [tex]\( y = mx + b \)[/tex] so it's easier to graph.
Starting with:
[tex]\[ x - 2y \geq -12 \][/tex]
Subtract [tex]\( x \)[/tex] from both sides:
[tex]\[ -2y \geq -x - 12 \][/tex]
Divide every term by [tex]\(-2\)[/tex]. Remember, dividing by a negative number reverses the inequality symbol:
[tex]\[ y \leq \frac{1}{2}x + 6 \][/tex]
### Step 2: Graph the Boundary Line
Now, graph the line [tex]\( y = \frac{1}{2}x + 6 \)[/tex].
1. Find the y-intercept: When [tex]\( x = 0 \)[/tex],
[tex]\[ y = 6 \][/tex]
Plot the point (0, 6) on the graph.
2. Find another point: Substitute another value for [tex]\( x \)[/tex]. Let's use [tex]\( x = 2 \)[/tex],
[tex]\[ y = \frac{1}{2}(2) + 6 = 1 + 6 = 7 \][/tex]
Plot the point (2, 7).
3. Plot the points and draw the line: Draw a straight line through these points. Since the original inequality includes "greater than or equal to" (≥), this line will be solid, indicating that points on the line satisfy the inequality.
### Step 3: Shade the Appropriate Region
The inequality is [tex]\( y \leq \frac{1}{2}x + 6 \)[/tex], meaning we need to shade the region below the line where [tex]\( y \)[/tex] is less than or equal to the value given by the line for that [tex]\( x \)[/tex].
### Step 4: Verifying with a Test Point
To ensure correct shading, you can choose a test point not on the line, such as (0, 0).
Substitute [tex]\( x = 0 \)[/tex] and [tex]\( y = 0 \)[/tex] into the original inequality:
[tex]\[ 0 - 2(0) \geq -12 \][/tex]
[tex]\[ 0 \geq -12 \][/tex]
This is true, which means that the point (0, 0) satisfies the inequality and should be in the shaded region. Hence, the region that includes (0, 0) is correctly shaded below the line.
### Final Graph
- Draw a solid line for [tex]\( y = \frac{1}{2}x + 6 \)[/tex].
- Shade the region below this line.
This represents the graph of the inequality [tex]\( x - 2y \geq -12 \)[/tex].
### Step 1: Rewrite the Inequality in Slope-Intercept Form
First, rewrite the inequality in the form [tex]\( y = mx + b \)[/tex] so it's easier to graph.
Starting with:
[tex]\[ x - 2y \geq -12 \][/tex]
Subtract [tex]\( x \)[/tex] from both sides:
[tex]\[ -2y \geq -x - 12 \][/tex]
Divide every term by [tex]\(-2\)[/tex]. Remember, dividing by a negative number reverses the inequality symbol:
[tex]\[ y \leq \frac{1}{2}x + 6 \][/tex]
### Step 2: Graph the Boundary Line
Now, graph the line [tex]\( y = \frac{1}{2}x + 6 \)[/tex].
1. Find the y-intercept: When [tex]\( x = 0 \)[/tex],
[tex]\[ y = 6 \][/tex]
Plot the point (0, 6) on the graph.
2. Find another point: Substitute another value for [tex]\( x \)[/tex]. Let's use [tex]\( x = 2 \)[/tex],
[tex]\[ y = \frac{1}{2}(2) + 6 = 1 + 6 = 7 \][/tex]
Plot the point (2, 7).
3. Plot the points and draw the line: Draw a straight line through these points. Since the original inequality includes "greater than or equal to" (≥), this line will be solid, indicating that points on the line satisfy the inequality.
### Step 3: Shade the Appropriate Region
The inequality is [tex]\( y \leq \frac{1}{2}x + 6 \)[/tex], meaning we need to shade the region below the line where [tex]\( y \)[/tex] is less than or equal to the value given by the line for that [tex]\( x \)[/tex].
### Step 4: Verifying with a Test Point
To ensure correct shading, you can choose a test point not on the line, such as (0, 0).
Substitute [tex]\( x = 0 \)[/tex] and [tex]\( y = 0 \)[/tex] into the original inequality:
[tex]\[ 0 - 2(0) \geq -12 \][/tex]
[tex]\[ 0 \geq -12 \][/tex]
This is true, which means that the point (0, 0) satisfies the inequality and should be in the shaded region. Hence, the region that includes (0, 0) is correctly shaded below the line.
### Final Graph
- Draw a solid line for [tex]\( y = \frac{1}{2}x + 6 \)[/tex].
- Shade the region below this line.
This represents the graph of the inequality [tex]\( x - 2y \geq -12 \)[/tex].
