6-3B Solving Multi-Step Inequalities

Algebra 1 Glencoe McGraw-Hill Linda Stamper

A multi-step inequality is solved by transforming the inequality more than one time. Undo addition or subtraction before undoing multiplication or division or you may make the problem more complicated to solve. Always remember the basic rule when isolating the variable:

Solving Multi–Step Inequalities

Whatever you do to one side of the

inequality sign, you must also do to the

other side of the inequality sign.

To solve inequalities that have variables on both sides of the inequality symbol, first collect the variable terms on one side.

Solve .

Write the problem.Collect variable terms on one side.

Undo addition and simplify.

Undo multiplication and simplify.

31x322x6 x6 x6

31x322 31 31

x39 3 3

x3

31x322x6

3x

To solve inequalities that have variables on both sides of the inequality symbol, first collect the variable terms on one side.

Solve .

Write the problem.Collect variable terms on one side.

Undo addition and simplify.

Undo multiplication and simplify.

31x322x6 x3 x3

3122x3

22 229x3

3 33x

31x322x6

</

When solving inequalities that contain grouping symbols, first use the distributive property.

Write problem. 2x3x33x6 Distribute and combine like terms.

6x3x318x6 6x318x9

Collect variable terms on one side.

x3x36 18x6

Isolate the variable using inverse operations.

18 1824x6

6 6 4x

.2x3x33x6Solve

Solve.

w6 w1264 w12w12w1864

81 81

w932

Leave the fraction

improper. It must be in lowest

terms.

Example 1 Example 255x219x6

x2 x25519x4

19 1936x4

4 49x

Example 315x9x216x5

x7 x715x216

15 15x21

2 2

x21

15x916x7

Use good form –

place the negative out front!

932

w

>/

21

x

If solving an inequality results in a statement that is always true, the solution is the set of all real numbers. If solving an inequality results in a statement that is never true, the solution is an empty set, written as .

Write problem.

Distribute and combine like terms.

Collect variable terms on one side.

x5x5

Result is a false statement.2728

.87x54x32x8Solve

Copy in your spiral notebook!

87x54x32x8

835x512x316x8 27x528x5

Write problem.

Distribute and combine like terms.

Collect variable terms on one side.

x2x2

Result is a true statement.

2424

.12x212x4x3Solve

Copy in your spiral notebook!

all real numbers

12x212x4x3 24x212x12x3 24x224x2

Write problem.

Distribute and combine like terms.

Collect variable terms on one side.

2424

Result is a true statement.

x2x2

.12x212x4x3Solve

all real numbers

12x212x4x3 24x212x12x3 24x224x2

Subtract 24 from each side.

Be sure to simplify the inequality to determine whether it is a true statement. If the inequality is true, the solution is the set of all real numbers. If the inequality is false, the solution is the empty set, written as .

xx

Solve.

2b323 1)

6 68 b9

9

98

b

29b6

>/

9

7m9 81m 2) m m

7m8 81 77

m8 88 8 8

m 11 11 m

</

54

x

10 10 810x 2 2

6210x 3x 3x

6x32x7

Solve.

6x5 x415x8 3)

x5 x56 15x7

151512 x7 7 7 3 x

6x5 15x12

2x327x 4)

Solve.

1x22x8x114x7 5)

x4 x4228

2x4x8x1128x7 2x428x4

Solve.

y4 y4

183

12y2y3y23y4 6)

12y6y3y23y4

18y43y4

all real numbers

7) One half of a number increased by one is greater than or equal to six less than one fourth of the number.

6n41

1n21

n41n

41

1 1

6n41

1n42

61n41

7n41

14

14

28n

Write an inequality and then solve.

8) Sixteen more than three fourths of a number is more than one eighth of the number less than 2.

n81

216n43

n81n

81

16 16

n81

216

216n87

14n87

78

78

16n

-2

Write an inequality and then solve.

n43

n86

6-A4 Pages 311–312 #26–33, 39–42, 44–47.