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Company | Membership | Blog | Help center | User guides | Tell us what you think | Testimonials | Careers | Contact us | Terms of service | Privacy policy © 2022 IXL Learning. All rights reserved. Follow us Learning Objectives By the end of this section, you will be able to: Prerequisite Skills Before you get started, take this prerequisite quiz. 1. Is \((3, 12)\) a solution to \(y>2x+3\)? Click here to check your answerYes, because \(12>9\). If you missed this problem, review Section 4.1. (Note that this will open in a new window.) 2. Graph all solutions to \(2x-3y<12\). Click here to check your answerIf you missed this problem, review Section 4.1. (Note that this will open in a new window.) 3. Where do the lines \(y=2x+1\) and \(y=-3x+6\) intersect? Click here to check your answer\((1, 3)\) If you missed this problem, review Section 3.1. (Note that this will open in a new window.) Determine whether an ordered pair is a solution of a system of linear inequalitiesThe definition of a system of linear inequalities is very similar to the definition of a system of linear equations. SYSTEM OF LINEAR INEQUALITIES Two or more linear inequalities grouped together form a system of linear inequalities. A system of linear inequalities looks like a system of linear equations, but it has inequalities instead of equations. A system of two linear inequalities is shown here. \[\left\{\begin{array} {l} x+4y\geq 10\\3x−2y<12\end{array}\right.\nonumber\] To solve a system of linear inequalities, we will find values of the variables that are solutions to both inequalities. We solve the system by using the graphs of each inequality and show the solution as a graph. We will find the region on the plane that contains all ordered pairs \((x,y)\) that make both inequalities true. SOLUTIONS OF A SYSTEM OF LINEAR INEQUALITIES Solutions of a system of linear inequalities are the values of the variables that make all the inequalities true. The solution of a system of linear inequalities is shown as a shaded region in the x, y coordinate system that includes all the points whose ordered pairs make the inequalities true. To determine if an ordered pair is a solution to a system of two inequalities, we substitute the values of the variables into each inequality. If the ordered pair makes both inequalities true, it is a solution to the system. Example \(\PageIndex{1}\) Determine whether the ordered pair is a solution to the system \(\left\{\begin{array} {l} x+4y\geq 10\\3x−2y<12\end{array}\right.\) a. \((−2,4)\) b. \((3,1)\) Answera. Is the ordered pair \((−2,4)\) a solution? The ordered pair \((−2,4)\) made both inequalities true. Therefore \((−2,4)\) is a solution to this system. b. Is the ordered pair \((3,1)\) a solution? The ordered pair \((3,1)\) made one inequality true, but the other one false. Therefore \((3,1)\) is not a solution to this system. Example \(\PageIndex{2}\) Determine whether the ordered pair is a solution to the system: \(\left\{ \begin{array} {l} x−5y>10\\2x+3y>−2 \end{array} \right.\) a. \((3,−1)\) b. \((6,−3)\) Answera. no b. yes Example \(\PageIndex{3}\) Determine whether the ordered pair is a solution to the system: \(\left\{ \begin{array} {l} y>4x−2\\4x−y<20 \end{array} \right.\) a. \((−2,1)\) b. \((4,−1)\) Answera. yes b. no Solve a System of Linear Inequalities by GraphingThe solution to a single linear inequality is the region on one side of the boundary line that contains all the points that make the inequality true. The solution to a system of two linear inequalities is a region that contains the solutions to both inequalities. To find this region, we will graph each inequality separately and then locate the region where they are both true. The solution is always shown as a graph. SOLVE A SYSTEM OF LINEAR INEQUALITIES BY GRAPHING.
Example \(\PageIndex{4}\): How to Solve a System of Linear Inequalities by Graphing Solve the system by graphing: \(\left\{\begin{array} {l} y\geq 2x−1 \\ y<x+1\end{array}\right.\) Solution Example \(\PageIndex{5}\) Solve the system by graphing: \(\left\{\begin{array} {l} y<3x+2\\y>−x−1\end{array}\right.\) AnswerThe solution is the darkest shaded region. Example \(\PageIndex{6}\) Solve the system by graphing: \(\left\{\begin{array} {l} y<−\dfrac{1}{2}x+3 \\ y<3x−4\end{array}\right.\) AnswerThe solution is the darkest shaded region. Example \(\PageIndex{7}\) Solve the system by graphing: \(\left\{\begin{array} {l} x−y>3\\y<−15x+4\end{array}\right.\) Answer
The point of intersection of the two lines is not included as both boundary lines were dashed. The solution is the area shaded twice—which appears as the darkest shaded region. Example \(\PageIndex{8}\) Solve the system by graphing: \(\left\{\begin{array} {l} x+y\leq 2 \\ y\geq \frac{2}{3}x−1\end{array}\right.\) AnswerThe solution is the darkest shaded region. Example \(\PageIndex{9}\) Solve the system by graphing: \(\left\{\begin{array} {l} 3x−2y\leq 6\\y>−\frac{1}{4}x+5\end{array} \right.\) AnswerThe solution is the darkest shaded region. Example \(\PageIndex{10}\) Solve the system by graphing: \(\left\{\begin{array} {l} x−2y<5\\y>−4\end{array}\right.\) Answer
The point \((0,0)\) is in the solution and we have already found it to be a solution of each inequality. The point of intersection of the two lines is not included as both boundary lines were dashed. The solution is the area shaded twice—which appears as the darkest shaded region. Example \(\PageIndex{11}\) Solve the system by graphing: \(\left\{\begin{array} {l} y\geq 3x−2 \\ y<−1\end{array}\right.\) AnswerThe solution is the darkest shaded region. Example \(\PageIndex{12}\) Solve the system by graphing: \(\left\{\begin{array} {l} x>−4x−2 \\ y\geq −4 \end{array}\right.\) AnswerThe solution is the darkest shaded region. Systems of linear inequalities where the boundary lines are parallel might have no solution. We’ll see this in the next example. Example \(\PageIndex{13}\) Solve the system by graphing: \(\left\{\begin{array} {l} 4x+3y\geq 12 \\ y<−\frac{4}{3}x+1\end{array}\right.\) Answer
There is no point in both shaded regions, so the system has no solution. Example \(\PageIndex{14}\) Solve the system by graphing: \(\left\{\begin{array} {l} 3x−2y\geq 12 \\ y\geq \frac{3}{2}x+1\end{array}\right.\) AnswerNo solution. Example \(\PageIndex{15}\) Solve the system by graphing: \(\left\{\begin{array} {l} x+3y>8\\y<−\frac{1}{3}x−2\end{array}\right.\) AnswerNo solution. Some systems of linear inequalities where the boundary lines are parallel will have a solution. We’ll see this in the next example. Example \(\PageIndex{16}\) Solve the system by graphing: \(\left\{\begin{array} {l} y>\frac{1}{2}x−4\\x−2y<−4\end{array}\right.\) Answer
No point on the boundary lines is included in the solution as both lines are dashed. The solution is the region that is shaded twice which is also the solution to \(x−2y<−4\). Example \(\PageIndex{17}\) Solve the system by graphing: \(\left\{\begin{array} {l} y\geq 3x+1 \\ −3x+y\geq −4\end{array}\right.\) AnswerThe solution is the darkest shaded region.
Example \(\PageIndex{18}\) Solve the system by graphing: \(\left\{\begin{array} {l} y\leq −\frac{1}{4}x+2\\x+4y\leq 4\end{array}\right.\) AnswerThe solution is the darkest shaded region. Solve Applications of Systems of InequalitiesThe first thing we’ll need to do to solve applications of systems of inequalities is to translate each condition into an inequality. Then we graph the system, as we did above, to see the region that contains the solutions. Many situations will be realistic only if both variables are positive, so we add inequalities to the system as additional requirements. Example \(\PageIndex{19}\) Christy sells her photographs at a booth at a street fair. At the start of the day, she wants to have at least 25 photos to display at her booth. Each small photo she displays costs her $4 and each large photo costs her $10. She doesn’t want to spend more than $200 on photos to display. a. Write a system of inequalities
to model this situation. a. To find the system of equations translate the information. \( \qquad \begin{array} {l} \\ \\ \text{She wants to have at least 25 photos.} \\ \text{The number of small plus the number of large should be at least }25. \\ \hspace{45mm} x+y\geq 25 \\ \\ \\ $4 \text{ for each small and }$10\text{ for each large must be no more than }$200 \\ \hspace{40mm} 4x+10y\leq 200 \\ \\ \\ \text{The number of small photos must be greater than or equal to }0. \\ \hspace{50mm} x\geq 0 \\ \\ \\ \text{The number of large photos must be greater than or equal to }0. \\ \hspace{50mm} y\geq 0 \end{array} \) We have our system of equations. \(\hspace{65mm} \left\{\begin{array} {l} x+y\geq 25 \\4x+10y\leq 200\\x\geq 0\\y\geq 0\end{array}\right.\) b.
The solution of the system is the region of the graph that is shaded the darkest. The boundary line sections that border the darkly-shaded section are included in the solution as are the points on the x-axis from (25, 0) to (55, 0). c. To determine if 10 small and 20 large photos would work, we look at the graph to see if the point (10, 20) is in the solution region. We could also test the point to see if it is a solution of both equations. It is not, Christy would not display 10 small and 20 large photos. d. To determine if 20 small and 10 large photos would work, we look at the graph to see if the point (20, 10) is in the solution region. We could also test the point to see if it is a solution of both equations. It is, so Christy could choose to display 20 small and 10 large photos. Notice that we could also test the possible solutions by substituting the values into each inequality. Example \(\PageIndex{20}\) A trailer can carry a maximum weight of 160 pounds and a maximum volume of 15 cubic feet. A microwave oven weighs 30 pounds and has 2 cubic feet of volume, while a printer weighs 20 pounds and has 3 cubic feet of space. a. Write a system of inequalities to model this situation. a. \(\left\{\begin{array} {l} 30m+20p\leq 160\\2m+3p\leq 15\end{array}\right.\) c. yes Example \(\PageIndex{21}\) Mary needs to purchase supplies of answer sheets and pencils for a standardized test to be given to the juniors at her high school. The number of the answer sheets needed is at least 5 more than the number of pencils. The pencils cost $2 and the answer sheets cost $1. Mary’s budget for these supplies allows for a maximum cost of $400. a. Write a system of inequalities to model this situation. a. \(\left\{\begin{array} {l} a\geq p+5 \\ a+2p\leq 400\end{array}\right.\) c. no When we use variables other than x and y to define an unknown quantity, we must change the names of the axes of the graph as well. Example \(\PageIndex{22}\) Omar needs to eat at least 800 calories before going to his team practice. All he wants is hamburgers and cookies, and he doesn’t want to spend more than $5. At the hamburger restaurant near his college, each hamburger has 240 calories and costs $1.40. Each cookie has 160 calories and costs $0.50. a. Write a system of inequalities to model this situation. a. To find the system of equations translate the information. The calories from hamburgers at 240 calories each, plus the calories from cookies at 160 calories each must be more that 800. \(\qquad \begin{array} {l} \hspace{40mm} 240h+160c\geq 800 \\ \\ \\ \text{The amount spent on hamburgers at }$1.40\text{ each, plus the amount spent on cookies}\\\text{at }$0.50\text{ each must be no more than }$5.00.\\ \hspace{40mm} 1.40h+0.50c\leq 5 \\ \\ \\ \text{The number of hamburgers must be greater than or equal to 0.} \\ \hspace{50mm} h\geq 0 \\ \text{The number of cookies must be greater than or equal to 0.}\\ \hspace{50mm} c\geq 0 \end{array} \) \(\text{We have our system of equations.} \qquad \left\{ \begin{array} {l} 240h+160c\geq 800 \\ 1.40h+0.50c\leq 5 \\ h\geq 0 \\ c\geq 0\end{array} \right.\) b.
Graph \(1.40h+0.50c\leq 5\). The boundary line is \(1.40h+0.50c=5\). We test (0, 0) and it makes the inequality true. We shade the side of the line that includes (0, 0). The solution of the system is the region of the graph that is shaded the darkest. The boundary line sections that border the darkly shaded section are included in the solution as are the points on the x-axis from (5, 0) to (10, 0). c. To determine if 3 hamburgers and 2 cookies would meet Omar’s criteria, we see if the point (3, 2) is in the solution region. It is, so Omar might choose to eat 3 hamburgers and 2 cookies. d. To determine if 2 hamburgers and 4 cookies would meet Omar’s criteria, we see if the point (2, 4) is in the solution region. It is, Omar might choose to eat 2 hamburgers and 4 cookies. We could also test the possible solutions by substituting the values into each inequality. Example \(\PageIndex{23}\) Tension needs to eat at least an extra 1,000 calories a day to prepare for running a marathon. He has only $25 to spend on the extra food he needs and will spend it on $0.75 donuts which have 360 calories each and $2 energy drinks which have 110 calories. a. Write a system of inequalities that models this situation. a. \(\left\{\begin{array} {l} 0.75d+2e\leq 25\\360d+110e\geq 1000\end{array}\right.\) c. yes Example \(\PageIndex{24}\) Philip’s doctor tells him he should add at least 1,000 more calories per day to his usual diet. Philip wants to buy protein bars that cost $1.80 each and have 140 calories and juice that costs $1.25 per bottle and have 125 calories. He doesn’t want to spend more than $12. a. Write a system of inequalities that models this situation. a. \(\left\{\begin{array} {l} 140p+125j\geq 1000\\1.80p+1.25j\leq 12\end{array}\right.\) c. yes Access these online resources for additional instruction and practice with solving systems of linear inequalities by graphing.
Key Concepts
Glossarysystem of linear inequalitiesTwo or more linear inequalities grouped together form a system of linear inequalities.How do you solve systems of inequalities by graphing?Solve a System of Linear Inequalities by Graphing. Graph the first inequality. Graph the boundary line. ... . On the same grid, graph the second inequality. Graph the boundary line. ... . The solution is the region where the shading overlaps.. Check by choosing a test point.. How do you solve systems of inequalities?Step 1: Solve the inequality for y. ... . Step 2: Graph the boundary line for the inequality. ... . Step 3: Shade the region that satisfies the inequality. ... . Step 4: Solve the second inequality for y. ... . Step 5: Graph the boundary line for the second inequality. ... . Step 6: Shade the region that satisfies the second inequality.. What is system of inequalities in math?A system of inequalities is a set of two or more inequalities in one or more variables. Systems of inequalities are used when a problem requires a range of solutions, and there is more than one constraint on those solutions.
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