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Chapter 7: Problem 40

Write each sentence as an inequality in two variables. Then graph the inequality. The \(y\)-variable is at least 2 more than the product of \(-3\) and the \(x\)-variable.

Short Answer

Expert verified
The inequality is \(y \geq -3x + 2\). When graphed, it is a downward sloping line with a solid line and shading above it.

Step by step solution

01

Writing the Inequality

Begin by determining the relationship between \(y\) and \(x\). The sentence 'The \(y\)-variable is at least 2 more than the product of -3 and the \(x\)-variable' needs to be translated to an inequality. This can be written as \(y \geq -3x + 2\).
02

Graphing the Inequality

Then, the inequality is graphed on a coordinate plane. Considering the line \(y = -3x + 2\), plot a point on the y-intercept (0, 2). Then, since the slope is -3, move down 3 units and one unit to the right to plot the second point. Repeat the process to plot additional points as needed. Once the points are plotted, connect them with a straight line to form the graph. Since the inequality has a 'greater than or equal to' sign, use a solid line to denote that every point on the line also satisfies the inequality.
03

Shading the Graph

For \(y \geq -3x + 2\), shade the region above the line because the inequality is 'greater than or equal to'. This shaded region represents all the possible pairs of coordinates (x, y) that satisfy the inequality.

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Most popular questions from this chapter

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. \(y=-x^{2}+2 x+3\)

Describe how to solve a system of linear inequalities.

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