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Chapter 8: Problem 50

Graph the solution set of each inequality. $$6 x-2 y \leq 24$$

Short Answer

Expert verified
The graph of the inequality \(6x - 2y \leq 24\) is a solid line passing through the point (0, -12) and extending to the 1st and 3rd quadrants. The area above this line (including the line) is shaded to represent the solution to the inequality.

Step by step solution

01

Transform inequality to y-intercept form

To easily graph the inequality, transform it into slope-intercept form \(y = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept. In this case, divide the whole inequality by 2 to isolate y: \(6x - 2y \leq 24 \Rightarrow 3x - y \leq 12 \Rightarrow y \geq 3x - 12\). So the coefficient of \(x\) (3) is the slope of the line and -12 is the y-intercept.
02

Draw the boundary line

Plot the line by first marking the y-intercept (-12) on the y-axis. From this point, rise in the direction determined by the slope (3), this means move 3 units up from the y-intercept and 1 unit to the right. Continue these steps to plot more points. Since the original inequality includes the equal sign (\(\leq\)), the boundary line should be solid, meaning it is part of the solution set.
03

Determine which half-plane to shade

The inequality is \(y \geq 3x - 12\), so the portion of the graph above the line (including the line) is the solution to the inequality. Choose a test point not on the line (0,0 is a common choice if it's not on the line) to confirm this. Substitute these coordinates into our inequality: \(0 \geq 3(0) - 12\) which is false, meaning the region containing (0,0) is not in the solution set. So, we shade the half-plane that does not include the origin.
04

Check your graph

Any coordinate point located in the shaded region or on the line should make the inequality true. Test a few points in the shaded area, for instance, (4,3). Substituting into the inequality, we get \(3 \geq 3(4) - 12\) which simplifies to \(3 \geq 0\). This is a true statement, which verifies that our graph is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope-Intercept Form
The slope-intercept form is a way of expressing linear equations as \( y = mx + b \). In this form:
  • \( m \) represents the slope of the line.
  • \( b \) represents the y-intercept.
Converting inequalities into the slope-intercept form helps in graphing because it gives us immediate information about the slope and where the line crosses the y-axis.
For example, let鈥檚 take the inequality \( 6x - 2y \leq 24 \). To put it into slope-intercept form, solve for \( y \) by isolating it on one side of the inequality:
  • First, rearrange the terms: \( 3x - y \leq 12 \).
  • Then, solve for \( y \): \( y \geq 3x - 12 \).
Here, the slope \( m \) is 3, and the y-intercept \( b \) is -12.
This tells us the line will rise 3 units for every 1 unit it runs to the right, starting from the point where the line crosses the y-axis at -12.
Boundary Line
When graphing an inequality, the boundary line is the line that corresponds to the equation version of the inequality. For \( y \geq 3x - 12 \), the boundary line is represented by the equation \( y = 3x - 12 \).
To graph this line:
  • Start by plotting the y-intercept \(-12\) on the y-axis.
  • Use the slope \( m = 3 \) to determine the next point. From \(-12\), rise 3 units up and move 1 unit right.
  • Repeat to find more points if needed, and then draw the line through these points.
Since the inequality \( y \geq 3x - 12 \) includes "\( \leq \)," the boundary line is solid to indicate that points on the line satisfy the inequality. Each point on this line makes the inequality true.
Half-Plane Shading
To show the solution set of an inequality on a graph, we shade the region that satisfies the inequality. In our example, the inequality is \( y \geq 3x - 12 \).
Here鈥檚 how to determine which half-plane to shade:
  • Take a test point not on the line, such as (0,0).
  • Substitute the coordinates into the inequality \( 0 \geq 3(0) - 12 \).
  • The statement \( 0 \geq -12 \) is false, meaning the area containing (0,0) is not part of the solution set.
  • Therefore, shade the half-plane on the side of the line not containing the origin.
Always choose a test point to confirm which side of the boundary line to shade. The area you shade represents all the solutions to the inequality, and any point in this region will satisfy the inequality \( y \geq 3x - 12 \).

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Most popular questions from this chapter

Consider the following system of equations. $$\left\\{\begin{array}{l} y=x^{2}+6 x-4 \\ y=b \end{array}\right.$$ For what value(s) of \(b\) do the graphs of the equations in this system have (a) exactly one point of intersection? (b) exactly two points of intersection? (c) no point of intersection?

An electronics firm makes a clock radio in two different models: one (model 380 ) with a battery backup feature and the other (model 360 ) without. It takes 1 hour and 15 minutes to manufacture each unit of the model 380 radio, and only 1 hour to manufacture each unit of the model \(360 .\) At least 500 units of the model 360 radio are to be produced. The manufacturer realizes a profit per radio of \(\$ 15\) for the model 380 and only \(\$ 10\) for the model \(360 .\) If at most 2000 hours are to be allocated to the manufacture of the two models combined, how many of each model should be made to maximize the total profit?

A chemist wishes to make 10 gallons of a \(15 \%\) acid solution by mixing a \(10 \%\) acid solution with a \(25 \%\) acid solution. (a) Let \(x\) and \(y\) denote the total volumes (in gallons) of the \(10 \%\) and \(25 \%\) solutions, respectively. Using the variables \(x\) and \(y,\) write an equation for the total volume of the \(15 \%\) solution (the mixture). (b) Using the variables \(x\) and \(y,\) write an equation for the total volume of acid in the mixture by noting that Volume of acid in \(15 \%\) solution \(=\) volume of acid in \(10 \%\) solution \(+\) volume of acid in \(25 \%\) solution. (c) Solve the system of equations from parts (a) and (b), and interpret your solution. (d) Is it possible to obtain a \(5 \%\) acid solution by mixing a \(10 \%\) solution with a \(25 \%\) solution? Explain without solving any equations.

A farmer has 90 acres available for planting corn and soybeans. The cost of seed per acre is \(\$ 4\) for corn and \(\$ 6\) for soybeans. To harvest the crops, the farmer will need to hire some temporary help. It will cost the farmer \(\$ 20\) per acre to harvest the corn and \(\$ 10\) per acre to harvest the soybeans. The farmer has \(\$ 480\) available for seed and \(\$ 1400\) available for labor. His profit is \(\$ 120\) per acre of corn and \(\$ 150\) per acre of soybeans. How many acres of each crop should the farmer plant to maximize the profit?

Apply elementary row operations to a matrix to solve the system of equations. If there is no solution, state that the system is inconsistent. $$\left\\{\begin{aligned} x+3 y &=2 \\ 5 x+12 y+3 z &=1 \\\\-4 x-9 y-3 z &=1 \end{aligned}\right.$$
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