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4.5 Systems of Linear Inequalities

Linear Inequalities in Two Variables

Examples of linear inequalities in two variables: 4x+2y 12, y 2x 3

Linear Inequalities in two variables are of the form Ax+By > C or Ax+By < C, where A, B,and C are real numbers. (Combined linear equality and inequality statements are of the form Ax+By C or Ax+By C.)

The graph of the solutions of a linear inequality in two variables is a half-plane.

3. Pick any ordered pair that is not on the graphed line. This will be your test point. Substitute the coordinates of that ordered pair into the original inequality. If the inequality is true, shade the side of the line where the test point is. If the inequality is false, shade the other side of the line.

Next Slide

1. Rewrite the linear inequality as a linear equation. Change the inequality symbol ( < , > , , or ) to an equal sign.

2. The next step is to graph that line. If the inequality symbol has an equal sign ( or ), draw a solid line. If the inequality symbol does not have an equal sign (< , >), draw a dashed line.

Procedure for Graphing Linear Inequalities

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4.5 Systems of Linear Inequalities

Write the inequality as a linear equation, then graph.

x y

0 3

2 03(0) 2(0)<6 0<6

x axis

y axis

Pick a test point not on the line, say (0,0).

Since this is true, we can shade the side where the test point is.

Example 1. Graph: 3x +2y < 6

Solution:

3x + 2y = 6

Graph 4x 3y 12

Your Turn Problem #1

Answer:

x y

0 -4 3 0

y axis

x axis

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4.5 Systems of Linear Inequalities

Your Turn Problem #2

1Graph y x

2

Write the inequality as a linear equation, and graph.

x y0 01 22 4

x axis

y axis

y 2x 4 2(0) 4 0

Since this is false, we must shade the side opposite the test point.

Note: There is another method to determining which side of the line to shade. If the inequality is y> or y≥, shade above the line. If the inequality is y< or y≤, shade below the line. This example is y>2x, therefore we shade above the line. Using this method takes a little work to get y by itself but then you don’t have to worry about the test point.

Example 2. Graph y > 2x

Solution:

y = 2x Pick a test point, say (0,4).

y axis

x axis

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4.5 Systems of Linear Inequalities

Your Turn Problem #3

Graph: y 4

x = 4

x axis

y axis

Pick a test point, (0, 0)

0 4

x 4 is a vertical line thatintersects the x axis at 4.(solid since )

Since this is false, shade on the opposite side of the test point.

Note: If the inequality is x> or x≥, shade to the right of the line. If the inequality is x< or x≤, shade to the left of the line. This example is x≤−4, therefore we shade to the left of the line.

GrExam aph ple 3. x 4Solution:

x axis

y axis

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4.5 Systems of Linear Inequalities

Your Turn Problem #4

3Graph: y x 2

4

x y0 23 06 -2

x axis

y axis

Because our inequality is y >, shade above the dashed line.

2y x 2

3

1st, write the inequality as linear equation to graph the line. Either use the y intercept and the slope or choose values for x. If you choose values for x, always choose x=0 and also choose values that are multiples of the denominator in the fraction.

2 Graph Example 4. y x 2

3

x axis

y axis

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4.5 Systems of Linear Inequalities

Your Turn Problem #5

Graph x > 3 and y < 2

Answer

Recall that the word “and” indicates the intersection of the solutions sets of each inequality.

x 2 is a vertical line through (2,0). x axis

y axis

The shaded area will be to the left.

y 1 is a horizontal line through 1.

The shaded area will be below the line.

The intersection is the area that satisfied both inequalities. (shaded twice)

Graph x 2Example and y 5 .. 1

Solution:

x axis

y axis

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4.5 Systems of Linear Inequalities

In the previous sections, we solved systems of equations such as:x y 52x y 4

The solution set of the system is the intersection of the solution sets of the individual lines. (i.e., where the two lines meet.)

The previous example contained the word “and”, which means the intersection. Another way of asking for the intersection is using a system such as:x y 5

2x y 4

The solution set of the system is the intersection of the solution sets of the individual inequalities. (i.e., where the graph is shaded twice.)

Next Slide

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4.5 Systems of Linear Inequalities

Graph x y 5.

y x 5

x y

0 5

1 4

2 3

y x 5

Since y >, shade above.

Graph 2x y 4.

Since y ≥, shade above.

The solution is the area which was shaded twice. We will darken that area with another color to show the intersection more clearly.

Next Slide

x y 52x y 4

Example 6. Solve the following system by graphing.

Solution:

x axis

y axis

x y

0 -4

1 -2

2 0

y 2x 4

y 2x 4

y 2x 4

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4.5 Systems of Linear Inequalities

x axis

y axis

Answer:

x y 0x 2y 2

Your Turn Problem #6.Solve the following system by graphing.

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4.5 Systems of Linear Inequalities

x 0y 03x 6y 185x 2y 10

Example 7. Solve the following system by graphing.

Solution:

Easier to graph last two inequalities using the intercept method. Then use test points to determine shaded area.

The solution is the area in Quadrant I which satisfies all four inequalities, shown here in the light blue color.

Next Slide

x 0 ; shade to the right of the y axis(or the line x=0).

y 0 ; shade above the x axis(or the line y=0).

Therefore, our graph will only be shaded in Quadrant I, the upper right-hand quadrant.

x axis

y axis

3x 6y 18

x y

0 3

6 0

5x 2y 10

x y

0 5

2 0

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4.5 Systems of Linear Inequalities

Your Turn Problem #7.

Graph: x 0y 06x 4y 244x 7y 28

x axis

y axis

(4,0)

(0,4)

(7,0)

(0,6)

Answer:

The End.B.R.1-25-07