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To solve and graph the inequality [tex]\(2x - 6y \geq 36\)[/tex] with a detailed, step-by-step approach, let's break down the problem into manageable parts:
### Step 1: Rewrite the Inequality in Slope-Intercept Form
First, let's rewrite the inequality in the form of [tex]\(y \leq mx + b\)[/tex] or [tex]\(y > mx + b\)[/tex]. Start by manipulating the given inequality.
[tex]\[2x - 6y \geq 36\][/tex]
Isolate [tex]\(y\)[/tex] by first subtracting [tex]\(2x\)[/tex] from both sides:
[tex]\[-6y \geq -2x + 36\][/tex]
Next, divide everything by [tex]\(-6\)[/tex]. Remember, when dividing by a negative number, the inequality sign flips:
[tex]\[y \leq \frac{-2x + 36}{-6}\][/tex]
[tex]\[y \leq \frac{2}{6}x - \frac{36}{6}\][/tex]
[tex]\[y \leq \frac{1}{3}x - 6\][/tex]
### Step 2: Determine the Boundary Line
The boundary line for our inequality is given by the equation:
[tex]\[y = \frac{1}{3}x - 6\][/tex]
This equation represents a straight line. To plot this line, we'll use some key points:
1. Y-Intercept: When [tex]\(x = 0\)[/tex], [tex]\(y = -6\)[/tex]. So, the point is [tex]\((0, -6)\)[/tex].
2. X-Intercept: When [tex]\(y = 0\)[/tex], solving for [tex]\(x\)[/tex]:
[tex]\[0 = \frac{1}{3}x - 6\][/tex]
[tex]\[6 = \frac{1}{3}x\][/tex]
[tex]\[x = 18\][/tex]
So, the point is [tex]\((18, 0)\)[/tex].
### Step 3: Graph the Boundary Line
Plot the points [tex]\((0, -6)\)[/tex] and [tex]\((18, 0)\)[/tex] on the coordinate plane and draw a straight line through them. The line represents [tex]\(y = \frac{1}{3}x - 6\)[/tex]. Because the original inequality is [tex]\(\geq\)[/tex], draw a solid line to show that points on the line are included in the solution set.
### Step 4: Determine the Shaded Region
The inequality [tex]\(y \leq \frac{1}{3}x - 6\)[/tex] tells us to shade below the line. We can confirm this by using a test point. A convenient test point not on the line is the origin [tex]\((0,0)\)[/tex].
Substitute [tex]\(x = 0\)[/tex] and [tex]\(y = 0\)[/tex] into the inequality:
[tex]\[0 \leq \frac{1}{3}(0) - 6\][/tex]
[tex]\[0 \leq -6\][/tex]
This statement is false. Therefore, the origin is not in the shaded region. So we will shade the opposite side of the line (below the line).
### Summary
1. The boundary line is [tex]\(y = \frac{1}{3}x - 6\)[/tex].
2. It is a solid line.
3. The shaded region is below the line.
On a graph, your boundary line will pass through the points [tex]\((0, -6)\)[/tex] and [tex]\((18, 0)\)[/tex], and the area below this line represents all the solutions to the inequality [tex]\(2x - 6y \geq 36\)[/tex].
### Step 1: Rewrite the Inequality in Slope-Intercept Form
First, let's rewrite the inequality in the form of [tex]\(y \leq mx + b\)[/tex] or [tex]\(y > mx + b\)[/tex]. Start by manipulating the given inequality.
[tex]\[2x - 6y \geq 36\][/tex]
Isolate [tex]\(y\)[/tex] by first subtracting [tex]\(2x\)[/tex] from both sides:
[tex]\[-6y \geq -2x + 36\][/tex]
Next, divide everything by [tex]\(-6\)[/tex]. Remember, when dividing by a negative number, the inequality sign flips:
[tex]\[y \leq \frac{-2x + 36}{-6}\][/tex]
[tex]\[y \leq \frac{2}{6}x - \frac{36}{6}\][/tex]
[tex]\[y \leq \frac{1}{3}x - 6\][/tex]
### Step 2: Determine the Boundary Line
The boundary line for our inequality is given by the equation:
[tex]\[y = \frac{1}{3}x - 6\][/tex]
This equation represents a straight line. To plot this line, we'll use some key points:
1. Y-Intercept: When [tex]\(x = 0\)[/tex], [tex]\(y = -6\)[/tex]. So, the point is [tex]\((0, -6)\)[/tex].
2. X-Intercept: When [tex]\(y = 0\)[/tex], solving for [tex]\(x\)[/tex]:
[tex]\[0 = \frac{1}{3}x - 6\][/tex]
[tex]\[6 = \frac{1}{3}x\][/tex]
[tex]\[x = 18\][/tex]
So, the point is [tex]\((18, 0)\)[/tex].
### Step 3: Graph the Boundary Line
Plot the points [tex]\((0, -6)\)[/tex] and [tex]\((18, 0)\)[/tex] on the coordinate plane and draw a straight line through them. The line represents [tex]\(y = \frac{1}{3}x - 6\)[/tex]. Because the original inequality is [tex]\(\geq\)[/tex], draw a solid line to show that points on the line are included in the solution set.
### Step 4: Determine the Shaded Region
The inequality [tex]\(y \leq \frac{1}{3}x - 6\)[/tex] tells us to shade below the line. We can confirm this by using a test point. A convenient test point not on the line is the origin [tex]\((0,0)\)[/tex].
Substitute [tex]\(x = 0\)[/tex] and [tex]\(y = 0\)[/tex] into the inequality:
[tex]\[0 \leq \frac{1}{3}(0) - 6\][/tex]
[tex]\[0 \leq -6\][/tex]
This statement is false. Therefore, the origin is not in the shaded region. So we will shade the opposite side of the line (below the line).
### Summary
1. The boundary line is [tex]\(y = \frac{1}{3}x - 6\)[/tex].
2. It is a solid line.
3. The shaded region is below the line.
On a graph, your boundary line will pass through the points [tex]\((0, -6)\)[/tex] and [tex]\((18, 0)\)[/tex], and the area below this line represents all the solutions to the inequality [tex]\(2x - 6y \geq 36\)[/tex].
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