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Chapter 8: Problem 1

Graph each inequality. $$x+2 y \leq 8$$

Short Answer

Expert verified
First, isolate \(y\) to achieve \(y \leq -\frac{1}{2}x + 4\). Then graph the corresponding equation \(y = -\frac{1}{2}x + 4\) as a solid line because the inequality includes \(\leq\), which means values on the line also satisfy the inequality. The solution to the inequality \(y \leq -\frac{1}{2}x + 4\) is the region below this line.

Step by step solution

01

Isolate y

Firstly, isolate y in the inequality. The inequality \(x +2y \leq 8\) can be rearranged to isolate \(y\) by subtracting \(x\) from both sides and then dividing by \(2\). This gives us \(y \leq -\frac{1}{2}x + 4\).
02

Draw the Boundary Line

Now that \(y\) is isolated, take the corresponding equation \(y = -\frac{1}{2}x + 4\), and graph this on the coordinate plane. This line is the boundary of the area defined by the inequality. Since our original inequality is \(\leq\), the boundary line will be solid, because points on the line will be included in the solutions.
03

Identify the Solution Region

Finally, identify the solution region. Since the inequality was \(y \leq\), the solution region will be the area below the line. The final graph is a solid line for \(y = -\frac{1}{2}x + 4\) and the region below this line shaded.

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