Problem 51 - Beginning and Intermediate Algebra

Question

Graph each compound inequality. \(y \leq-x-1\) or \(x \geq 6\)

Step-by-Step Solution

Answer

To graph the compound inequality \(y \leq -x - 1\) or \(x \geq 6\), follow these steps: 1. Graph the boundary line \(y = -x - 1\) as a solid line. Shade the region below this line. 2. Graph the boundary line \(x=6\) as a solid vertical line. Shade the region to the right of this line. 3. On a single coordinate plane, include both shaded regions and the solid boundary lines as part of the solution.

Step 1: Graph \(y\leq -x - 1\)

To graph the inequality \(y \leq -x - 1\), we first graph its boundary line \(y = -x - 1\). This line divides the plane into two regions: one satisfies the inequality, and the other doesn't. Since the inequality is "less than or equal to," we know that the boundary line will be part of the solution, so it will be a solid line. To do this, find the two points for \(y = -x - 1\). When \(x = 0\), we have \(y = -1\). When \(y = 0\), we have \(x = -1\). Now we plot the points \((0, -1)\) and \((-1, 0)\) and draw a solid line connecting them. Next, we need to figure out which side of this line satisfies \(y \leq -x - 1\). Choose a test point, for example, the origin \((0, 0)\). Plug in these coordinates into the inequality: \(0 \leq -0 - 1\), which simplifies to \(0 \leq -1\). Since this is false, we know that the region *opposite* to the test point is the one that satisfies the inequality. So, shade the region below the line \(y = -x - 1\).

Step 2: Graph \(x \geq 6\)

To graph the inequality \(x \geq 6\), we first need to graph its boundary line \(x = 6\). Notice that this is a vertical line that passes through the x-coordinate 6, and similarly to the previous boundary, since the inequality sign is "greater than or equal to," we will use a solid line. Now, to find which side of this line satisfies the given inequality, let's consider a test point: for example, \((7,0)\). When we plug it into \(x \geq 6\), we have \(7 \geq 6\), which is true. So, the region that includes this test point, to the right of the line \(x=6\), satisfies the inequality. Consequently, we shade this region.

Step 3: Combine both shaded regions

Now that we have graphed and shaded the regions for both inequalities, we can plot them on the same coordinate plane and look for their combined regions. Because this is an "or" compound inequality, we are looking for any point that satisfies *either* of the inequalities, or both. In other words, plot both shaded regions on the same coordinate plane. The final solution includes the whole area that is covered by either of the shaded regions. The graph should show a shaded region below the line \(y = -x -1\) and another shaded region to the right of the line \(x=6\), with the solid lines being part of the solution as well.