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TOPIC 8

1. SOLUTION OF LINEAREQUALITY AND LINEAR INEQUALITY WITH ONE VARIABLE.

2. GRAPH INEQUALITY

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DEFINITION:

Equation

Inequality

Solution of an Equation

Two expressions set equal to each other

Linear EquationAn equation that can be written in the form  ax + b = 0 where a and b are constants

A value, such that, when you replace the variable with it, it makes the equation true. (the left side comes out equal to the right side)

Mathematical expressions that usethe symbols ( <, ≠, >, ≥, etc.)

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Solve  for x. 10 2x

Solving a Linear Equation  in General

Get the variable you are solving for alone on one side and everything else on the other side using INVERSE operations.

10 2x *Inverse of sub. 10 is add. 10

10 10 2 10x

8x

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Solve    for x.

*Inverse of add. 7 is sub. 7

5 7x 5 7x

5 7 7 7x

12 x

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*Inverse of div. by 3 is mult. by 3

Solve for x

73

x

73

x

3 3 73

x

21x

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Inverse of mult. by 5 is div. by 5

Solve   for x.5 4x

5 4x 5 4

5 5

x

4

5x

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Solve.

4x + 6 = x

Solving Equations with Variables on Both Sides

4x + 6 = x– 4x – 4x

6 = –3x

To collect the variable terms on one side, subtract 4x from both sides.

Since x is multiplied by -3, divide both sides by –3.

–2 = x

6–3

–3x–3

=

You can always check your solution by substituting the value back into the original equation.

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Solve.

3b – 2 = 2b + 12

3b – 2 = 2b + 12

– 2b – 2b

b – 2 = 12+ 2 + 2

b = 14

Since 2 is subtracted from b, add 2 to both sides.

To collect the variable terms on one side, subtract 2b from both sides.

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Solve.

3w + 1 = 3w + 8

1 ≠ 8

3w + 1 = 3w + 8– 3w – 3w To collect the variable terms on

one side, subtract 3w from both sides.

No solution. There is no number that can be substituted for the variable w to make the equation true.

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1. 4x + 16 = 2x 6. 2x + 8 = x – 7

2. 8x – 3 = 15 + 5 7. –4(x + 3) = –5x – 2x

3. 2(3x + 11) = 6x + 4 8. 5x + x + (–11) = 25 – 3x

4. 2x + 9x – 3x + 8 = 16 9.

5. –4 = 6x + 22 – 4x 10.

x = –8

no solution

x = 1

x = –13

x = –15

x = 4

x = 4

Solve.

2 4 3 2

3 2

x x

4 5

6 2 3 4x x x

2

5x

72

8x

14

15x

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SOLVING LINEAR INEQUALITY IN ONE

VARIABLE

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An inequality compares two expressions using <, >, , or .

Symbol Meaning Word Phrases

<

>

≤

≥

is less than

is greater than

is greater than or equal to

is less than or equal to

Fewer than, below

More than, above

At most, no more than

At least, no less than

An inequality that contains a variable is an algebraic inequality.

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A solution of an inequality is any value of the variable that makes the inequality true. All of the solutions of an inequality are called the solution set.

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Solve the inequality.

Solving Inequalities by Adding or Subtracting

x + 3 > –5

x + 3 > –5–3 –3

x > –8

Since 3 is added x, subtract 3 from both sides.

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m – 4 ≥ –2

m – 4 ≥ –2+ 4 +4

m ≥ 2

Since 4 is subtracted from m, add 4 to both sides.

Solve

Solving Inequalities by Adding or Subtracting

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When you multiply (or divide) both sides of an inequality by a

negative number, you must reverse the inequality symbol to make the statement true.

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b ≥ –5

–9b ≤ 45

Divide both sides by –9;

≤ changes to ≥.

Solve

Solving Inequalities by Multiplying or Dividing

≥ 45

–9

–9b

–9

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48 < a, or a > 48

12 <

Multiply both sides by 4.

Solve

Solving Inequalities by Multiplying or Dividing

a 44 • 12 < 4 • a

4

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80 > b, or b < 80

16 >

Multiply both sides by 5.

Solve

b 5

5 • 16 > 5 • b 5

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Solve the inequality.

x + 4 > –2

x + 4 > –2–4 –4

x > –6

Since 4 is added x, subtract 4 from both sides.

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GRAPHINGINEQUALITY IN ONE

VARIABLE

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You can graph the solution set on a number line. The symbols < and > indicate an open circle while ≥ and ≤ indicate a close circle.

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This open circle shows that 5 is not a solution.

a > 5

The symbols ≤ and ≥ indicate a closed circle.

This closed circle shows that 3 is a solution.

b ≤ 3

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Graph each inequality.

–3 –2 –1 0 1 2 3

A. –1 > y Draw an open circle at –1. The solutions are all values of y less than –1, so shade the line to the left of –1.

B. z ≥ –2

–3 –2 –1 0 1 2 3

Graphing Inequalities

12 Draw a closed circle

at –2 and all values of z greater than2 . So shade to the right of –2 .

1 2 1

2 1 2

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Graph each inequality.

–3 –2 –1 0 1 2 3

A. n < 3

B. a ≥ –4

–6 –4 –2 0 2 4 6

Example 3

Draw an open circle at 3. The solutions are all values of n less than 3, so shade the line to the left of 3.

Draw a closed circleat –4. The solutions are all values greater than –4, so shade to the right of –4.

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Solve and graph.1. –14x > 28

2. < 15

5

3. 18 < –6x

x < –2

q ≥ 40

–3 > x

x < 45

–2 0 2

5040 45

40 454.

x

3

q

8

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6. –3 < y

0º

1 2 3–1–23 –

Graph each inequality.

5. m ≤ 1

– – – 0 1 2 3123