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3. Systems of Linear Equations & Inequalities
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United States of AmericaTennessee
Integrated Math 1

3.04 Systems of linear inequalities

Lesson
Practice
Lesson

Concept summary

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.

Worked 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.

Solution

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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Reflection

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.

Approach

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.

Solution

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

b

Sketch a graph of the system of inequalities.

Solution

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

Solution

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.

Outcomes

M1.A.CED.A.3

Create individual and systems of equations and/or inequalities to represent constraints in a contextual situation, and interpret solutions as viable or non-viable.*

M1.A.REI.D.6

Graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

M1.MP1

Make sense of problems and persevere in solving them.

M1.MP2

Reason abstractly and quantitatively.

M1.MP3

Construct viable arguments and critique the reasoning of others.

M1.MP4

Model with mathematics.

M1.MP5

Use appropriate tools strategically.

M1.MP6

Attend to precision.

M1.MP7

Look for and make use of structure.

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