6 1 solving linear inequalities in one variable
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6.1 Solving Linear Inequalities in One Variable. A linear inequality i n one variable is an inequality which can be put into the form. a x + b > c. where a , b , and c are real numbers . Note that the “>” can be replaced by  , < , or  .

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6.1 Solving Linear Inequalities in One Variable

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6 1 solving linear inequalities in one variable

6.1 Solving Linear Inequalities in One Variable

A linear inequalityin onevariable is an inequality which can be put into the form

ax + b > c

where a, b, and c are real numbers.

Note that the “>” can be replaced by , <, or .

Examples: Linear inequalities in one variable.

2x+ 3> 4

can be written –4x+ (–2)<–5.

2x – 2 < 6x – 5

6x+ 1  3(x – 5)

can be written 6x+1–15.


6 1 solving linear inequalities in one variable1

0

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-1

-1

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4

6.1 Solving Linear Inequalities in One Variable

The solution set for an inequality can be expressed in two ways.

Example: Express the solution set of x <3 in two ways.

Open circle indicate that the number is not included in the solution set.

{x | x<3}

1. Set-builder notation:

°

2. Graph on the real line:

Example:Express the solution set of x 4 in two ways.

Closed circle indicate that the number is included in the solution set.

{x | x4}

1. Set-builder notation:

2. Graph on the real line:


6 1 solving linear inequalities in one variable2

6.1 Solving Linear Inequalities in One Variable

A solution of an inequality in one variable is a number which, when substituted for the variable, results in a true inequality.

Examples: Are any of the values of x given below solutions of 2x > 5?

?

?

x = 2

2(2) > 5

4 > 5

False

2 is not a solution.

?

?

x = 2.6

2(2.6) > 5

5.2 > 5

True

2.6 is a solution.

?

?

x = 3

2(3) > 5

6 > 5

True

3 is a solution.

?

?

x = 1.5

2(1.5) > 5

3 > 5

False

1.5 is not a solution.

The solution set of an inequality is the set of all solutions.


6 1 solving linear inequalities in one variable3

6.1 Solving Linear Inequalities in One Variable

The graph of a linear inequality in one variable is the graph on the real number line of all solutions of the inequality.

Verbal Phrase

Inequality

Graph

All real numbers less than 2

x < 2

All real numbers greater than -1

x > -1

All real numbers less than or

equal to 4

x< 4

All real numbers greater than or equal to -3

x> -3


6 1 solving linear inequalities in one variable4

+ 4

+ 4

6.1 Solving Linear Inequalities in One Variable

Addition and Subtraction Properties

  • If a > b and c is a real number, then a > b, a + c > b + c, and a – c > b – c have the same solution set.

  • If a < b and c is a real number, then a < b, a + c < b + c, and a – c < b – c have the same solution set.

Example: Solve x – 4 > 7.

Add 4 to each side of the inequality.

x– 4 > 7

x > 11

Set-builder notation.

{x | x > 11}


6 1 solving linear inequalities in one variable5

– 2x

– 2x

6.1 Solving Linear Inequalities in One Variable

Example: Solve 3x2x + 5.

Subtract 2x from each side.

3x2x + 5

x 5

{x | x5}

Set-builder notation.


6 1 solving linear inequalities in one variable6

6.1 Solving Linear Inequalities in One Variable

Multiplication and Division Properties

  • If c > 0 the inequalities a > b,ac > bc, and>have the same solution set.

  • If c < 0 the inequalities a > b,ac < bc, and<have the same solution set.

Example: Solve 4x 12.

4x 12

Divide by 4.

x  3

4 is greater than 0, so the inequality sign remains the same.


6 1 solving linear inequalities in one variable7

Example: Solve .

(–3)

(–3)

6.1 Solving Linear Inequalities in One Variable

Multiply by –3.

–3 is less than 0, so the inequality sign changes.


6 1 solving linear inequalities in one variable8

x >

6.1 Solving Linear Inequalities in One Variable

Example: Solve x + 5 < 9x + 1.

–8x + 5 < 1

Subtract 9x from both sides.

–8x < –4

Subtract 5 from both sides.

Divide both sides by –8 and simplify.

Inequality sign changes because of division by a negative number.

Solution set in set-builder notation.


6 1 solving linear inequalities in one variable9

Example: Solve .

Subtract from both sides.

-20

-10

0

10

20

30

40

50

60

6.1 Solving Linear Inequalities in One Variable

Add 2 to both sides.

Multiply both sides by 5.

Solution set as a graph.


6 2 problem solving

6.2 Problem Solving

Example: A cell phone company offers its customers a rate of $89 per month for 350 minutes, or a rate of $40 per month plus $0.50 for each minute used.

How many minutes per month can a customer who chooses the second plan use before the charges exceed those of the first plan?

Let x = the number of minutes used.

Solve the inequality 0.50x + 40  89.

0.50x 49

Subtract 40.

x  24.5

Divide by 0.5.

The customer can use up to 24.5 minutes per month before the cost of the second plan exceeds the cost of the first plan.


6 3 compound inequalities

6.3 Compound Inequalities

A compound inequality is formed by joining two inequalities with “and” or “or.”


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6.3 Compound Inequalities

Example: Solve x + 2 < 5 and 2x – 6 > – 8.

Solve the first inequality.

Solve the second inequality.

x +2 < 5

2x – 6 > – 8

x < 3

2x > – 2

Subtract 2.

Add 6.

{x | x < 3}

x > – 1

Divide by 2.

Solution set

{x | x > –1}

Solution set

The solution set of the “and” compound inequality is the intersection of the two solution sets.

°

º


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6.3 Compound Inequalities

When solving compound inequalities, it is possible to work with both inequalities at once.

This inequality means 11 < 6x + 5 and 6x + 5 < 29.

Example: Solve 11 < 6x + 5 < 29.

6 < 6x < 24

Subtract 5 from each of the three parts.

1 < x < 4

Divide 6 into each of the three parts.

Solution set.


6 3 compound inequalities3

Example: Solve .

0

-4

-3

-2

-1

1

2

3

4

6.3 Compound Inequalities

Subtract 6 from each part.

Multiply each part by –2.

Multiplication by a negative number changes the inequality sign for each part.

Solution set.


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6.3 Compound Inequalities

Example: Solve x + 5 > 6 or 2x < –4.

Solve the first inequality.

Solve the second inequality.

x + 5 > 6

2x < –4

x > 1

x < –2

{ x | x > 1}

{ x | x < –2}

Solution set

Solution set

Since the inequalities are joined by “or” the solution set is the union of the solution sets.

°

°


6 4 absolute value and inequalities

6.4 Absolute Value and Inequalities

There are 4 types of absolute value inequalities and equivalent inequalities

  • |x| < a

  • |x| <a

  • |x| > a

  • |x| >a


6 4 absolute value and inequalities1

6.4 Absolute Value and Inequalities

Translating Absolute Value Inequalities

  • The inequality |ax + b| < c is equivalent to -c < ax + b < c

  • The inequality |ax + b| > c is equivalent to ax + b < -c or

    ax + b > c


6 4 absolute value and inequalities2

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º º

6.4 Absolute Value and Inequalities

Example Solve |x - 4| < 3

  • -3 < x - 4 < 3

  • 1 < x < 7

  • The solution set is {x| 1 < x < 7} and the interval is (1, 7)


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

6.4 Absolute Value and Inequalities

Example Solve |4x - 1| < 9

  • -9 ≤ 4x - 1 ≤ 9

  • -8 ≤ 4x≤ 10

  • -2 ≤x≤ 5/2

  • The interval solution is [-2, 5/2]


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º º

6.4 Absolute Value and Inequalities

Solve |x| > a

  • Solve |x + 1| > 2

  • x + 1 < -2 or x + 1 > 2

  • x < -3 or x > 1

  • The solution interval is (-∞, -3) U (1, ∞)


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• •

6.4 Absolute Value and Inequalities

Solve |x| ≥a

  • Solve |2x - 8| ≥ 4

  • 2x – 8 ≤ -4 or 2x - 8 ≥ 4

  • 2x≤ 4 or 2x≥ 12

  • x≤ 2 or x≥ 6

  • The solution interval is (-∞, 2] U [ 6, ∞)


6 5 linear inequalities

6.5 Linear Inequalities

  • A linear inequality in two variables can be written in any one of these forms:

    • Ax + By < C

    • Ax + By > C

    • Ax + By ≤ C

    • Ax + By ≥ C

  • An ordered pair (x, y) is a solution of the linear inequality if the inequality is TRUE when x and y are substituted into the inequality.


6 5 linear inequalities1

6.5 Linear Inequalities

  • Which ordered pair is a solution of

    5x - 2y ≤ 6?

    • (0, -3)

    • (5, 5)

    • (1, -2)

    • (3, 3)


6 5 linear inequalities2

6.5 Linear Inequalities

  • The graph of a linear inequality is the set of all points in a coordinate plane that represent solutions of the inequality.

    • We represent the boundary line of the inequality by drawing the function represented in the inequality.


6 5 linear inequalities3

6.5 Linear Inequalities

  • The boundary line will be a:

    • Solid line when ≤ and ≥ are used.

    • Dashed line when < and > are used.

  • Our graph will be shaded on one side of the boundary line to show where the solutions of the inequality are located.


6 5 linear inequalities4

6.5 Linear Inequalities

Here are some steps to help graph linear inequalities:

  • Graph the boundary line for the inequality. Remember:

    • ≤ and ≥ will use a solid curve.

    • < and > will use a dashed curve.

  • Test a point NOT on the boundary line to determine which side of the line includes the solutions. (The origin is always an easy point to test, but make sure your line does not pass through the origin)

    • If your test point is a solution (makes a TRUE statement), shade THAT side of the boundary line.

    • If your test points is NOT a solution (makes a FALSE statement), shade the opposite side of the boundary line.


6 5 linear inequalities5

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x

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-5

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6.5 Linear Inequalities

  • Graph the inequality x ≤ 4 in a coordinate plane

  • Decide whether to

    use a solid or

    dashed line.

  • Use (0, 0) as a

    test point.

  • Shade where the

    solutions will be.


6 5 linear inequalities6

y

5

x

-5

-5

5

6.5 Linear Inequalities

  • Graph y>x + 2 in a coordinate plane.

  • Sketch the boundary line of the graph.

  • Solid or dashed

    line?

  • Use (0, 0) as a

    test point.

  • Shade where the

    solutions are.


6 5 linear inequalities7

y

5

x

-5

-5

5

6.5 Linear Inequalities

  • Graph y> -½x - 2 in a coordinate plane.

  • Sketch the boundary line of the graph.

  • Solid or dashed

    line?

  • Use (0, 0) as a

    test point.

  • Shade where the

    solutions are.


6 5 linear inequalities8

y

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x

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6.5 Linear Inequalities

  • Graph 3x - 4y > 12 in a coordinate plane.

  • Sketch the boundary line of the graph.

    • Find the x- and

      y-intercepts and

      plot them.

  • Solid or dashed

    line?

  • Use (0, 0) as a

    test point.

  • Shade where the

    solutions are.


Example 4 using a new test point

y

5

x

-5

-5

5

Example 4:Using a new Test Point

  • Graph y < 2/5x in a coordinate plane.

  • Sketch the boundary line of the graph.

    • Find the x- and y-intercept and plot them.

    • Both are the origin!

  • Use the line’s slope

    to graph another point.

  • Solid or dashed

    line?

  • Use a test point

    OTHER than the

    origin.

  • Shade where the

    solutions are.


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