Solve the system of two linear inequalities graphically.

$\left\{\begin{array}{l}
y \leq 6 x+9 \\
y>-2 x-3
\end{array}\right.$

Step 1 of 3: Graph the solution set of the first linear inequality.

Choose the type of boundary line:

To graph the inequality y ≤ 6 x + 9 , we find the boundary line y = 6 x + 9 and draw a solid line since the inequality includes points on the line. We then shade below this line, indicating the region where the inequality holds true. The graphical solution is all points below and including the line.
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Answered by Anonymous | 2025-07-04

Find the boundary line by replacing the inequality sign with an equality sign: y = 6 x + 9 .
Determine that the boundary line is solid because the inequality is y ≤ 6 x + 9 .
Find two points on the line, such as ( 0 , 9 ) and ( − 1 , 3 ) , and plot the line.
Determine that the region below the line should be shaded because the inequality is y ≤ 6 x + 9 .

Explanation

Finding the Boundary Line We are given the inequality y ≤ 6 x + 9 . The first step in graphing this inequality is to determine the equation of the boundary line. The boundary line is found by replacing the inequality sign with an equality sign. Thus, the boundary line is y = 6 x + 9 .

Identifying Slope and Intercept The boundary line is y = 6 x + 9 . This is a linear equation in slope-intercept form, y = m x + b , where m is the slope and b is the y-intercept. In this case, the slope is m = 6 and the y-intercept is b = 9 .

Determining the Type of Boundary Line Since the inequality is y ≤ 6 x + 9 , the boundary line is included in the solution. Therefore, the boundary line is solid.

Finding Two Points on the Line To graph the line, we need to find two points on the line. Let's choose x = 0 and x = − 1 . When x = 0 , y = 6 ( 0 ) + 9 = 9 . So the point ( 0 , 9 ) is on the line. When x = − 1 , y = 6 ( − 1 ) + 9 = − 6 + 9 = 3 . So the point ( − 1 , 3 ) is on the line.

Plotting the Line Now we plot the points ( 0 , 9 ) and ( − 1 , 3 ) and draw a solid line through them.

Determining the Shaded Region To determine which side of the line to shade, we can test a point that is not on the line. Let's test the point ( 0 , 0 ) . If we plug in x = 0 and y = 0 into the inequality y ≤ 6 x + 9 , we get 0 ≤ 6 ( 0 ) + 9 , which simplifies to 0 ≤ 9 . Since this is true, we shade the region below the line.

The Solution Set The solution set of the inequality y ≤ 6 x + 9 is the region below the solid line y = 6 x + 9 .


Examples
Linear inequalities are used in many real-world applications, such as determining the feasible region in linear programming problems. For example, a company might use linear inequalities to determine the optimal production levels of two different products, given constraints on resources such as labor and materials. By graphing the inequalities, the company can visualize the feasible region and find the production levels that maximize profit.

Answered by GinnyAnswer | 2025-07-04