Problem 47
Question
Graph each compound inequality.
\(y
Step-by-Step Solution
Verified Answer
The final graph will show the intersection of two shaded regions: below the dashed line \(y=x+4\) and above the solid line \(y=-3\). The solution region is bound by the dashed line for the \(y
1Step 1: Identify the type of inequalities
Both given inequalities represent linear equations. Notice the inequality symbols (< and ≥), which indicate the type of inequality each one is:
1. \(y < x + 4\) is a strict inequality, meaning we will use a dashed line to represent the border of the solution region.
2. \(y \geq -3\) is a non-strict inequality, meaning we will use a solid line to represent the border of the solution region.
2Step 2: Graph y = x + 4 as a dashed line
Start by graphing the line \(y = x + 4\). This line has a slope of 1 and a y-intercept of 4. Plot the y-intercept (0, 4) and use the slope to find other points on the line (e.g., (1, 5), (2, 6), (-1, 3), etc.). As this is a strict inequality, draw the line as a dashed line to indicate that the points on the line are not included in the solution region.
3Step 3: Shade the region where y < x + 4
Next, we need to identify the region where \(y < x + 4\). Since it's a less than sign, we shade the area below the dashed line, excluding the dashed line itself.
4Step 4: Graph y = -3 as a solid line
Now, graph the line \(y = -3\), which is a horizontal line. Since this is a non-strict inequality, draw the line as a solid line, indicating the points on the line are included in the solution region.
5Step 5: Shade the region where y ≥ -3
Finally, identify and shade the region in which \(y \geq -3\). Since it's a greater than or equal to symbol, we shade the area above and including the solid line.
6Step 6: Identify the intersection of the solution regions
The last step is to identify the intersection of the two shaded regions. The region where both inequalities are true simultaneously, also called the solution region to the compound inequality, is what we are looking for. This intersection should be a shaded region on the coordinate plane bound by the dashed line of \(y < x + 4\) and the solid line of \(y \geq -3\).
The final graph should show the intersection of the two shaded regions and display both the dashed and solid lines as relevant borders.
Other exercises in this chapter
Problem 46
Write an absolute value equation that means \(y\) is 6 units from zero.
View solution Problem 47
Explain why the solution to \(|2 x+1| \geq-3\) is \((-\infty, \infty)\).
View solution Problem 47
Write an absolute value equation that has a solution set of \(\left\\{-\frac{1}{2}, \frac{1}{2}\right\\}\)
View solution Problem 48
Explain why the solution to \(|7 y-3| \geq 0\) is \((-\infty, \infty)\).
View solution