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Chapter 7: Problem 12

Graph each linear inequality. \(y>\frac{1}{4} x\)

Short Answer

Expert verified
The graph starts with a dashed line of the function \(y=\frac{1}{4} x\), all points above the line are shaded to represent solutions to the inequality \(y>\frac{1}{4} x\).

Step by step solution

01

Graph the Line

First, a line of the function \(y=\frac{1}{4} x\) has to be sketched. Start with the y-intercept which is at 0 in this case and then go up one unit and over 4 units from there to plot the slope. Continue to do so and draw the line through the plotted points. The line should be dashed since the points on the line aren't included in the solution set of the inequality (as the inequality sign is '>' and not '>=').
02

Decide which area to shade

After drawing the boundary, it's time to decide which side of the line to shade. This inequality is \(y>\frac{1}{4} x\), which means we're concerned with y-values that are greater than one-fourth of their respective x-values, essentially the points above the line. So all points above the line are included in the solution set of the inequality. So the area above the line needs to be shaded.
03

Shade the appropriate area

Now that we have identified the area, shade it. This visualization shows all the possible solutions to the inequality \(y>\frac{1}{4} x\), which are all the points above the line.

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Most popular questions from this chapter

The graphs of solution sets of systems of inequalities involve finding the intersection of the solution sets of two or more inequalities. By contrast, in Exercises 43-44, you will be graphing the union of the solution sets of two inequalities. Graph the union of \(x-y \geq-1\) and \(5 x-2 y \leq 10\).

In Exercises 41-42, write the given sentences as a system of inequalities in two variables. Then graph the system. The sum of the \(x\)-variable and the \(y\)-variable is at most 4 . The \(y\)-variable added to the product of 3 and the \(x\)-variable does not exceed \(6 .\)

Make Sense? Determine whether each statement makes sense or does not make sense, and explain your reasoning. When graphing \(3 x-4 y<12\), it's not necessary for me to graph the linear equation \(3 x-4 y=12\) because the inequality contains a \(<\) symbol, in which equality is not included.

Make Sense? Determine whether each statement makes sense or does not make sense, and explain your reasoning. I graphed the solution set of \(y \geq x+2\) and \(x \geq 1\) without using test points.

Graph each linear inequality. \(y>\frac{1}{3} x\)
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