Graph the inequality y < (1/4)x - 8 on the axes below.
Understand the Problem
The question is asking us to graph the inequality y < (1/4)x - 8 on a coordinate plane. This requires understanding how to represent inequalities on a graph, including determining the line and shading the appropriate region.
Answer
Graph of the inequality is a dashed line \(y = \frac{1}{4}x - 8\) shaded below the line.
Answer for screen readers
The graph of the inequality (y < \frac{1}{4}x - 8) contains a dashed line representing the boundary (y = \frac{1}{4}x - 8) with shading below the line.
Steps to Solve
- Identify the boundary line First, we need to rewrite the inequality as an equation to find the boundary line. The equation is
$$ y = \frac{1}{4}x - 8 $$
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Graph the boundary line Next, plot the boundary line on the graph. Since the inequality is (y < \frac{1}{4}x - 8), we will use a dashed line to indicate that points on the line are not included.
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Determine the y-intercept and x-intercept To plot the line, find the y-intercept (where (x = 0)) and the x-intercept (where (y = 0)):
- Y-intercept: $$ y = \frac{1}{4}(0) - 8 = -8 ; \text{(point (0, -8))} $$
- X-intercept: Set (y = 0): $$ 0 = \frac{1}{4}x - 8 $$ Solving for (x): $$ \frac{1}{4}x = 8 $$ $$ x = 32 ; \text{(point (32, 0))} $$
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Shade the appropriate region Since the inequality is (y < \frac{1}{4}x - 8), shade the area below the line. This indicates all points where (y) is less than the boundary line.
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Label the graph (optional) Label your axes and ensure the line is dashed with the shaded region clearly indicating where the solution to the inequality exists.
The graph of the inequality (y < \frac{1}{4}x - 8) contains a dashed line representing the boundary (y = \frac{1}{4}x - 8) with shading below the line.
More Information
The graph represents all the points where the value of (y) is less than the linear expression on the right. The slope of the line is (1/4), which means that for every 4 units you move right along the x-axis, you move 1 unit up along the y-axis.
Tips
- Forgetting to use a dashed line for less-than (<) or greater-than (>) inequalities.
- Shading the wrong region (above instead of below the line for "less than").
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