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Chapter 1: Problem 10

Graph the inequality: \(\mathrm{y}<-(1 / 2) \mathrm{x}+3\).

Short Answer

Expert verified
To graph the inequality \(y < -\frac{1}{2}x + 3\), first graph the boundary line \(y = -\frac{1}{2}x + 3\) as a dashed line by plotting points (\(0, 3\)) and (\(2, 2\)). Determine the solution region by testing the origin (0,0) and finding that it satisfies the inequality. Shade the region containing the origin to represent the solution.

Step by step solution

01

Determine the equation of the boundary line

The boundary line corresponds to the equality \(y = -\frac{1}{2} x + 3\). This line divides the coordinate plane into two regions - one where the inequality is true and the other where it is false.
02

Graph the boundary line

To graph the boundary line, we can identify two points that lie on the line and connect them. We can choose any two x-values and find their corresponding y-values. Let's choose \(x = 0\): \(y = -\frac{1}{2} (0) + 3 = 0 + 3 = 3\) So, we have the point \((0, 3)\) on the boundary line. Now, let's choose \(x = 2\): \(y = -\frac{1}{2} (2) + 3 = -1 + 3 = 2\) So, we have another point, \((2, 2)\) on the boundary line. Now, plot these points and draw a dashed line passing through them (since the inequality is strict without an equal sign).
03

Test a point to find the solution region

Now we need to determine which side of the boundary line is our solution region. To do this, choose a test point that is not on the line. A good test point to use is the origin (0,0) since it's easy to work with. Checking if the origin satisfies the inequality: \(0 < -\frac{1}{2}(0) + 3 \implies 0 < 3\) Since this statement is true, the origin is in the solution region. Therefore, the region containing the origin (0,0) is the solution region.
04

Graph the solution region

Shade the solution region that contains the origin (0,0). The resulting graph should have the dashed boundary line and the shaded region representing the inequality \(y < -\frac{1}{2} x + 3\).

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