Solve The System Of Inequalities By Graphing.$\[ \begin{align} y &\leq -7x + 8 \\ y &\ \textless \ -x + 3 \end{align} \\]- Select A Line To Change It Between Solid And Dotted.- Select A Region To Shade It.

Feb 28, 2025 by ADMIN 210 views

Introduction

In mathematics, solving a system of inequalities involves finding the solution set that satisfies all the given inequalities. One of the methods to solve a system of inequalities is by graphing. Graphing involves plotting the lines of the inequalities on a coordinate plane and then shading the regions that satisfy the inequalities. In this article, we will learn how to solve a system of inequalities by graphing.

Understanding the Inequalities

Before we start graphing, let's understand the given inequalities. We have two inequalities:

$y \leq -7x + 8$
$y < -x + 3$

The first inequality is a linear inequality in the form of $y \leq mx + b$ , where $m$ is the slope and $b$ is the y-intercept. The second inequality is also a linear inequality in the form of $y < mx + b$ . We will graph these inequalities on a coordinate plane.

Graphing the Inequalities

To graph the inequalities, we need to find the x and y intercepts of the lines. The x-intercept is the point where the line crosses the x-axis, and the y-intercept is the point where the line crosses the y-axis.

For the first inequality, $y \leq -7x + 8$ , we can find the x-intercept by setting $y = 0$ and solving for $x$ .

0 \leq -7x + 8

7x \leq 8

x \leq \frac{8}{7}

So, the x-intercept is $\left(\frac{8}{7}, 0\right)$ .

To find the y-intercept, we can set $x = 0$ and solve for $y$ .

y \leq -7(0) + 8

y \leq 8

So, the y-intercept is $(0, 8)$ .

For the second inequality, $y < -x + 3$ , we can find the x-intercept by setting $y = 0$ and solving for $x$ .

0 < -x + 3

x < 3

So, the x-intercept is $(3, 0)$ .

To find the y-intercept, we can set $x = 0$ and solve for $y$ .

y < -0 + 3

y < 3

So, the y-intercept is $(0, 3)$ .

Plotting the Lines

Now that we have the x and y intercepts, we can plot the lines on a coordinate plane.

For the first inequality, $y \leq -7x + 8$ , we can plot the line by drawing a line through the points $\left(\frac{8}{7}, 0\right)$ and $(0, 8)$ .

For the second inequality, $y < -x + 3$ , we can plot the line by drawing a line through the points $(3, 0)$ and $(0, 3)$ .

Shading the Regions

Now that we have plotted the lines, we need to shade the regions that satisfy the inequalities.

For the first inequality, $y \leq -7x + 8$ , we need to shade the region below the line.

For the second inequality, $y < -x + 3$ , we need to shade the region below the line.

Finding the Solution Set

The solution set is the region that satisfies both inequalities. To find the solution set, we need to find the intersection of the two shaded regions.

The solution set is the region that is shaded below both lines.

Conclusion

In this article, we learned how to solve a system of inequalities by graphing. We graphed the lines of the inequalities on a coordinate plane and then shaded the regions that satisfy the inequalities. We found the solution set by finding the intersection of the two shaded regions. This method is useful for solving systems of linear inequalities.

Example

Let's consider an example to illustrate the concept.

Suppose we have the following system of inequalities:

$y \leq 2x + 1$
$y < x - 2$

We can graph these inequalities on a coordinate plane and then shade the regions that satisfy the inequalities.

The solution set is the region that is shaded below both lines.

Tips and Tricks

Here are some tips and tricks to keep in mind when solving systems of inequalities by graphing:

Make sure to plot the lines accurately.
Shade the regions correctly.
Find the intersection of the two shaded regions to find the solution set.
Use a ruler or a straightedge to draw the lines.
Use different colors to shade the regions.

Conclusion

Q: What is the first step in solving a system of inequalities by graphing?

A: The first step in solving a system of inequalities by graphing is to understand the given inequalities and identify the type of inequality (linear, quadratic, etc.).

Q: How do I graph a linear inequality?

A: To graph a linear inequality, you need to find the x and y intercepts of the line. The x-intercept is the point where the line crosses the x-axis, and the y-intercept is the point where the line crosses the y-axis. Then, plot the line on a coordinate plane and shade the region that satisfies the inequality.

Q: What is the difference between a solid line and a dotted line in graphing inequalities?

A: A solid line represents an equality, while a dotted line represents an inequality. When graphing inequalities, use a solid line for equalities and a dotted line for inequalities.

Q: How do I find the solution set of a system of inequalities?

A: To find the solution set of a system of inequalities, you need to find the intersection of the two shaded regions. The solution set is the region that is shaded below both lines.

Q: What are some common mistakes to avoid when graphing inequalities?

A: Some common mistakes to avoid when graphing inequalities include:

Plotting the lines incorrectly
Shading the regions incorrectly
Not finding the intersection of the two shaded regions
Not using a ruler or straightedge to draw the lines

Q: Can I use graphing software to solve systems of inequalities?

A: Yes, you can use graphing software to solve systems of inequalities. Graphing software can help you plot the lines and shade the regions accurately.

Q: How do I determine which inequality to graph first?

A: When graphing a system of inequalities, it's best to graph the inequality with the steeper slope first. This will help you avoid confusion and make it easier to find the intersection of the two shaded regions.

Q: Can I use this method to solve systems of quadratic inequalities?

A: Yes, you can use this method to solve systems of quadratic inequalities. However, you will need to graph the quadratic functions and then shade the regions that satisfy the inequalities.

Q: How do I know if the solution set is empty or not?

A: If the two shaded regions do not intersect, then the solution set is empty. If the two shaded regions intersect, then the solution set is not empty.

Q: Can I use this method to solve systems of inequalities with multiple variables?

A: Yes, you can use this method to solve systems of inequalities with multiple variables. However, you will need to graph the functions in multiple dimensions and then shade the regions that satisfy the inequalities.

Conclusion

Solving a system of inequalities by graphing is a useful method for finding the solution set. By understanding the given inequalities, identifying the type of inequality, and graphing the lines, you can find the solution set. This method is useful for solving systems of linear inequalities and can be extended to solve systems of quadratic inequalities and inequalities with multiple variables.