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Chapter 8: Problem 2

Graph each inequality. $$3 x-6 y \leq 12$$

Short Answer

Expert verified
The inequality will be graphed with a line passing through the points (0,-2)... etc, with all points on and above the line shaded to represent the solution set to the inequality.

Step by step solution

01

Rearrange to Slope-Intercept Form

Start by rearranging the inequality into slope-intercept form, \(y = mx + b\). This involves isolating \(y\) on one side of the equation. So, our inequality would transform to \(y \geq \frac{1}{2}x - 2\).
02

Identify the Slope and Y-Intercept

In the inequality \(y \geq \frac{1}{2}x - 2\), the slope (m) is \(\frac{1}{2}\) and the y-intercept (b) is -2. The y-intercept is the point where the line crosses the y-axis and the slope is the steepness of the line. A positive slope means the line is ascending from left to right, while a negative slope means it's descending.
03

Plot the Y-Intercept and Graph the Line

First, plot the y-intercept on your graph, which is the point \(0, -2\). Since the slope is \(\frac{1}{2}\), you'd rise 1 unit and run 2 units to the right from the y-intercept to plot the next point. Draw a line through these points. This forms the boundary line of the solution set.
04

Determine Which Region to Shade

As the inequality is \(y \geq \frac{1}{2}x - 2\), you will be shading above the line. This indicates that any point in this region, including points on the line (since the inequality allows for \(y\) to equal \(\frac{1}{2}x - 2\)), is a solution to the inequality.

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Most popular questions from this chapter

determine whether each statement makes sense or does not make sense, and explain your reasoning. Use an extension of the Great Question! on page 859 to describe how to set up the partial fraction decomposition of a rational expression that contains powers of a prime cubic factor in the denominator. Give an example of such a decomposition.

An objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of \(x\) and \(y\) for which the maximum occurs. Objective Function Constraints $$ \begin{aligned} &z=5 x-2 y\\\ &\left\\{\begin{array}{l} {0 \leq x \leq 5} \\ {0 \leq y \leq 3} \\ {x+y \geq 2} \end{array}\right. \end{aligned} $$

An objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of \(x\) and \(y\) for which the maximum occurs. Objective Function Constraints $$ \begin{aligned} &z=10 x+12 y\\\ &\left\\{\begin{array}{l} {x \geq 0, y \geq 0} \\ {x+y \leq 7} \\ {2 x+y \leq 10} \\ {2 x+3 y \leq 18} \end{array}\right. \end{aligned} $$

Write the linear system whose solution set is \(\varnothing .\) Express each equation in the system in slope-intercept form.

Use the two steps for solving a linear programming problem, given in the box on page \(888,\) to solve the problems. A manufacturer produces two models of mountain bicycles. The times (in hours) required for assembling and painting each model are given in the following table: $$\begin{array}{lll} {} & {\text { Model } A} & {\text { Model } B} \\ {\text { Assembling }} & {5} & {4} \\ {\text { Painting }} & {2} & {3} \end{array}$$ The maximum total weekly hours available in the assembly department and the paint department are 200 hours and 108 hours, respectively. The profits per unit are 25 for model A and 15 for model B . How many of each type should be produced to maximize profit?
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