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Problem 10

# Find the graphical solution of each inequality. $$5 x-3 y \geq 15$$

Expert verified
The graphical solution of the inequality $$5x - 3y \geq 15$$ is the shaded region above the line $$y = \frac{5}{3}x - 5$$, including the line itself.
See the step by step solution

## Step 1: Convert the inequality into an equation

To graph the inequality, first, replace the inequality sign, $$\geq$$, with an equal sign, $$=$$, to get the equation: $$5x - 3y = 15$$

## Step 2: Find the slope and y-intercept

Now, we need to find the slope and the y-intercept of the equation. To find the slope, $$m$$, and the y-intercept, $$b$$, we need to rewrite the equation in slope-intercept form: $$y = mx + b$$. Let's solve for y: $$y = \frac{5}{3}x - 5$$ The slope, $$m$$, is $$\frac{5}{3}$$, and the y-intercept, $$b$$, is $$-5$$.

## Step 3: Graph the line on the Cartesian plane

Using the slope $$\frac{5}{3}$$ and y-intercept $$-5$$, we can graph the line on the Cartesian plane. 1. Plot the y-intercept at point $$(0, -5)$$. 2. From the y-intercept, use the slope (rise over run) to find another point on the line. In this case, we can go up 5 units and then right 3 units to arrive at point $$(3, 0)$$. Plot this point as well. 3. Connect the points with a straight solid line.

## Step 4: Determine the shaded region based on the inequality

Since the inequality is: $$5x - 3y \geq 15$$ The solution will be above the line we have graphed. Now, we must shade the region above the line (including the line itself) to represent the solution.

## Step 5: Identify the solution region

The graphical solution of the inequality $$5x - 3y \geq 15$$ is the shaded region above the line $$y = \frac{5}{3}x - 5$$. The line itself is also part of the solution, as points on the line satisfy the inequality.

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