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Chapter 3: Problem 2

On different grids, graph and shade in the areas described by the following linear inequalities. a. \(x-y<0\) b. \(x \leq-2\) c. \(3 x+2 y>6\) d. \(5 x-2 y \leq 10\)

Short Answer

Expert verified
Graph each inequality's boundary as a line (dashed or solid based on the inequality). Test a point to determine which side of the line to shade. Repeat for each inequality.

Step by step solution

01

Graph the Inequality

For each inequality, rewrite it in the form suitable for graphing. For instance, rewrite the inequality as an equation (equality) to find the boundary line. Use dashed lines for '<' and '>', and solid lines for 'ewline' . a. The inequality to graph is: .Step 1: Rewrite the inequality as an equation:ewline 1
02

Graph the Line

Graph the boundary line on the grid. For example, for part (a):ewline a. Graph the line:ewline .
03

Shading the Region

Determine which side of the line to shade by testing a point that is not on the line. If the point satisfies the inequality, shade that side of the line, otherwise shade the other side. For example, for part (a): a. Test the point (0,0):ewline In . Since it Another
04

Repeat for All Inequalities

Perform steps 1-3 for each inequality given: b. Inequality Step . Test.. . Shis. d. Inequality:. Step. Tes. .

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Key Concepts

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

linear inequalities
Linear inequalities involve expressions where two linear functions are compared using inequality signs like <, >, <=, or >=. Essentially, they indicate a range of values rather than a single value. For example, consider the inequality \( x - y < 0 \). It means that the difference between x and y should always be less than zero. Linear inequalities often describe regions on a coordinate plane rather than just points. To solve them, we typically find the boundary line first by changing the inequality to an equation (making it equal) and then determine which side of the line satisfies the inequality by testing points.
graphing
To graph a linear inequality, we first need to graph the 'boundary line', which is found by converting the inequality to an equation. For example, to graph \( x - y < 0 \), we start by plotting the line \( x - y = 0 \). Plot points where the equation holds true and draw the line. For inequalities involving '<' or '>', we use a dashed line to show that points on the line are not included in the solution. For '\( \leq \)' or '\( \geq \)', we use a solid line to include the boundary points.
shading regions
Once the boundary line is drawn, we need to determine which side of the line to shade. This shaded area represents all the solutions to the inequality. To do this, choose a test point not on the boundary line (often the origin, \((0,0)\) is a good choice if it's not on the line). Substitute this point into the original inequality to see if the inequality holds true. If the inequality is true for the test point, shade the side of the line where the test point lies. Otherwise, shade the opposite side. Example: For \( x - y < 0 \), test the point (0,0): \( 0 - 0 < 0 \), which is false, hence shade the region opposite to the origin.
boundary lines
Boundary lines divide the graph into two regions, one of which will satisfy the inequality. These lines are derived from the linear equation obtained by equating the inequality. A dashed boundary line is used when the inequality is strict ('<' or '>') indicating that points on the line are not included in the solution set. A solid boundary line is used for inclusive inequalities ('\( \leq \)' or '\( \geq \)'). When graphing boundary lines, it's essential to label them clearly and determine the correct line type to accurately represent the solution set.

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Most popular questions from this chapter

Predict the number of solutions to each of the following systems. Give reasons for your answer. You don't need to find any actual solutions. a. \(y=20,000+700 x \quad y=15,000+800 x\) b. \(y=20,000+700 x \quad y=15,000+700 x\) c. \(y=20,000+700 x \quad y=20,000+800 x\)

(Graphing program recommended.) The Ontario Association of Sport and Exercise Sciences recommends the minimum and maximum pulse rates \(P\) during aerobic activities, based on age \(A\). The maximum recommended rate, \(P_{\max },\) is \(0.87(220-A)\). The minimum recommended pulse rate, \(P_{\min },\) is \(0.72(220-A)\) a. Convert these formulas to the \(y=m x+b\) form. b. Graph the formulas for ages 20 to 80 years. Label the regions of the graph that represent too high a pulse rate, the recommended pulse rate, and too low a pulse rate. c. What is the maximum pulse rate recommended for a 20-year-old? The minimum for an 80 -year-old? d. Construct an inequality that describes too low a pulse rate for effective aerobic activity. e. Construct an inequality that describes the recommended pulse range.

a. Solve the following system algebraically: $$ \begin{array}{l} S=20,000+2500 n \\ S=25,000+2000 n \end{array} $$ b. Graph the system in part (a) and use the graph to estimate the solution to the system. Check your estimate with your answer in part (a).

You are thinking about replacing your long-distance telephone service. A cell phone company charges a monthly fee of \(\$ 40\) for the first 450 minutes and then charges \(\$ 0.45\) for every minute after \(450 .\) Every call is considered a longdistance call. Your local phone company charges you a fee of \(\$ 60\) per month and then \(\$ 0.05\) per minute for every longdistance call. a. Assume you will be making only long-distance calls. Create two functions, \(C(x)\) for the cell phone plan and \(L(x)\) for the local telephone plan, where \(x\) is the number of long-distance minutes. After how many minutes would the two plans cost the same amount? b. Describe when it is more advantageous to use your cell phone for long- distance calls. c. Describe when it is more advantageous to use your local phone company to make long-distance calls.

Solve the following systems: $$ \begin{array}{rrr} \text { a. } \frac{x}{3}+\frac{y}{2}=1 & \text { b. } \frac{x}{4}+y=9 \\ x-y=\frac{4}{3} & y=\frac{x}{2} \end{array} $$
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