Problem 61 - Beginning and Intermediate Algebra

Question

Graph each compound inequality. \(2 x+5 y<15\) or \(y \leq \frac{3}{4} x-1\)

Step-by-Step Solution

Answer

To graph the compound inequality \(2x + 5y < 15\) or \(y \leq \frac{3}{4}x - 1\), follow these steps: 1. Graph the boundary lines \(2x + 5y = 15\) (dashed line) and \(y = \frac{3}{4}x - 1\) (solid line) on the coordinate plane. 2. Determine the regions satisfying each inequality by using a test point, such as (0,0). 3. Shade the area that satisfies the first inequality and the area that satisfies the second inequality. 4. The final graph shows the shaded regions that satisfy either inequality, as they are connected by 'or'.

Step 1: Graph the first inequality: \(2x + 5y < 15\)

In order to graph the inequality, first change the inequality to an equation by replacing the "<" symbol with an equal sign: \[2x + 5y = 15\] Now we can find the x and y intercepts of this line to plot it on the coordinate plane. To find the x-intercept, set y = 0 and solve for x: \[ 2x + 5(0) = 15 \\ 2x = 15 \\ x = \frac{15}{2} \\ \] To find the y-intercept, set x = 0 and solve for y: \[ 2(0) + 5y = 15 \\ 5y = 15 \\ y = 3 \\ \] Now we have our x-intercept at \(\left(\frac{15}{2}, 0\right)\), and our y-intercept at \(\left(0,3\right)\). Let's plot these points and draw the boundary line. Since the original inequality is strictly less than (\(<\)), we draw a dashed line to represent the boundary.

Step 2: Shade the area satisfying the first inequality

To find which side of the boundary line to shade, we can choose any test point that doesn't lie on the line itself. A good test point to pick is the origin (0,0), because it makes calculations easier. Plug the test point (0,0) into the first inequality: \[2(0) + 5(0) < 15 \Rightarrow 0 < 15\] Since the test point satisfies the inequality, we can shade the entire area on the same side as the origin.

Step 3: Graph the second inequality: \(y \leq \frac{3}{4}x - 1\)

Similar to the first inequality, we change the inequality sign to an equal sign and find the x and y intercepts for the equation: \[ y = \frac{3}{4}x - 1 \] To find the x-intercept, set y = 0 and solve for x: \[ 0 = \frac{3}{4}x -1 \\ 1 = \frac{3}{4}x \\ x = \frac{4}{3} \] To find the y-intercept, set x = 0 and solve for y: \[ y =\frac{3}{4}(0) - 1 \\ y = -1 \] Now we have our x-intercept at \(\left(\frac{4}{3}, 0\right)\), and our y-intercept at \(\left(0,-1\right)\). Let's plot these points and draw the boundary line. Since the original inequality has a "less than or equal to" (\(\le\)) symbol, we draw a solid line to represent the boundary.

Step 4: Shade the area satisfying the second inequality

As before, use the test point (0,0) and plug it into the second inequality: \[0 \leq \frac{3}{4}(0) -1 \Rightarrow 0\leq -1\] Since the test point does not satisfy the inequality, we should shade the area on the opposite side of the line from the origin.

Step 5: Combine the shaded areas

Now that we have both inequalities graphed and their respective shaded regions, we need to combine them. Since the compound inequality is connected by "or", we should include both shaded regions in our final graph. The final graph will display the shaded regions that satisfy either inequality.