Topics
3. Linear equations and inequalities
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Australia
Year 10

3.04 Applications of linear inequalities

Lesson
Worksheet
Practice
Lesson

Introduction

A system of inequalities looks like a system of equations, but it has inequalities instead of equations.

To solve a system of inequalities, we will find values of the variables that are solutions to all the inequalities. We solve the system by using the graphs of each inequality and show the solution as a shaded region in the graph.

Applications of linear inequalities

A system of inequalities is a set of inequalities which have the same variables.

The solution to a system of inequalities is the set containing any ordered pair that makes all of the inequalities in the system true.

A solution can also be represented graphically as the region of the plane of the plane that satisfies all inequalities in the system.

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\begin{cases} y < x \\y\leq- 1 \end{cases}
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\begin{cases} y\geq x \\y\geq -x \end{cases}

The solution to a system of inequalities in a given context is viable if the solution makes sense in the context, and is non-viable if it does not make sense.

Examples

Example 1

Consider the following system of inequalities:

\begin{cases} y\leq 3 \\y > 4 x + 5\end{cases}

Sketch a graph of the solution set to the system of inequalities.

Worked Solution
Create a strategy

Graph each inequality at a time in the same coordinate plane and shade the region that satisfies both inequalities.

Apply the idea

To sketch the system of inequalities we can first construct the boundary lines for each inequality, namely y=3 and y=4x+5. When given a strict inequality we will draw a dashed line. When given a nonstrict inequality we will draw a solid line.

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To determine which side of each inequality will be shaded, we can choose some test points that satisfy each inequality. The test points will indicate which side of the inequality will be shaded. For y \leq 3 we will plot (-2,2) and (4,0). For y < 4x+5 we will plot (-2,5) and (-3,-3).

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The region that will be shaded is the region which satisfies both inequalities. Using the test points we can see that the shading will occur below the boundary line for y \leq 3 and to the left of y<4x+5.

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Reflect and check

Since each inequality in the system was already written in terms of y, it would have been possible to determine the direction of the shading without first plotting the test points. With some systems of inequalities written in terms of x or in general form, however, it can be less intuitive to know which direction to shade.

Example 2

Applicants for a particular university are asked to sit a quantitative reasoning test and verbal reasoning test. Successful applicants must obtain a minimum score of 14 on a quantitative reasoning test and a minimum combined score of 29 for both tests.

a

Write a system of inequalities for this scenario, where x represents the quantitative reasoning test score and y represents the verbal reasoning test score.

Worked Solution
Create a strategy

Since we know that the minimum accepted score for quantitative reasoning is 14, we can represent this with an inequality showing 14 as the lowest possible solution. A minimum combined score of 29 means the total of the two scores must sum to 29 or more.

Apply the idea

\begin{cases} x \geq 14 \\ x+y \geq 29 \end{cases}

b

Sketch a graph of the system of inequalities.

Worked Solution
Create a strategy

Graph each inequality at a time in the same coordinate plane and shade the region that satisfies both inequalities.

Apply the idea
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Does the solution (15,22.\overline{2}) make sense in terms of the context? Explain your answer.

Worked Solution
Apply the idea

No. This would mean that the score for the quantitative reasoning test was 15 and the verbal reasoning test 22.\overline{2}. By viewing the graph we can see that this point technically satsfies both inequalities, but a test score is typically a positive integer value or a simple fraction such as \dfrac{1}{2} or \dfrac{1}{4} and not a nonterminating decimal.

Idea summary

A system of inequalities is a set of two or more inequalities in the same variables.

The solution of a system of inequalities is the value of the variables that make all the inequalities true. It is shown as a shaded region in the Cartesian coordinate system and includes all the points whose ordered pairs make the inequalities true.

Outcomes

AC9M10A02

solve linear inequalities and simultaneous linear equations in 2 variables; interpret solutions graphically and communicate solutions in terms of the situation

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