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Chapter 7: Problem 13

Graph each linear inequality. \(y \leq 3 x+2\)

Short Answer

Expert verified
The graph of the inequality \(y \leq 3x + 2\) is the line itself (which is solid, to show it is included) and the area beneath the line.

Step by step solution

01

Sketch the Line

Start by graphing the related line \(y = 3x + 2\). To do this, plot the y-intercept (0,2). The slope is 3, which means moving 3 units up for every 1 unit to the right from the y-intercept point. Draw a line through the plotted points.
02

Fill in the Correct Half-Plane

The inequality \(y \leq 3x + 2\) means 'y' is less than or equal to \(3x + 2\), so the solutions are all of the points on the line and everywhere below it (as 'y' is less). Fill in all the area below the line to represent the solution set to the inequality.
03

Include the Line in the Solution

Since the inequality includes an 'or equal to' portion (\(\leq\)), the points on the line itself are also part of the solution to the inequality. Make sure the line in your graph is solid, not dashed, to indicate that it is included in the solution.

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Most popular questions from this chapter

What is a half-plane?

Compare the graphs of \(3 x-2 y>6\) and \(3 x-2 y \leq 6\). Discuss similarities and differences between the graphs.

Graph the solution set of each system of inequalities. \(\left\\{\begin{array}{l}4 x-5 y \geq-20 \\ x \geq-3\end{array}\right.\)

a. Create a scatter plot for the data in each table. b. Use the shape of the scatter plot to determine if the data are best modeled by a linear function, an exponential function, a logarithmic function, or a quadratic function. $$ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 4 \\ \hline 1 & 5 \\ \hline 2 & 7 \\ \hline 3 & 11 \\ \hline 4 & 19 \\ \hline \end{array} $$

a. Determine if the parabola whose equation is given opens upward or downward. b. Find the vertex. c. Find the \(x\)-intercepts. d. Find the \(y\)-intercept. e. Use (a)-(d) to graph the quadratic function. \(y=-x^{2}+2 x+3\)
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