The Modulus Function

Objectives:Understand what an absolute

value is.Solve equations and inequalities involving

modulus.Graph modulus functions.

The Modulus FunctionSUMMAR

YTo solve inequalities involving

one or more modulus functions:

Sketch the functions. ( Type “abs” if a graphical calculator is used. )

Find the points of intersection using the related functions ( without the mod. signs ).( A negative sign is needed for reflected parts. ) Refer to the graph to write the inequality for x.

Mark the section(s) of the x-axis that give the solution.

Method 1:

The Modulus FunctionSUMMAR

YMethod 2: Square both sides of the inequality to

remove the mod. signs and solve the resulting linear or quadratic inequality.( A graph is needed for quadratic

inequalities. )

The Modulus Function

1.

3 1

3 xy

1 xyx

y

Exercise

Solutions:

Solve the inequalities.

1. 31 xx 2. 12 x

The Modulus Function

1.

3 1

3 xy

1 xyx

y

Exercise

Solutions:

Solve the inequalities.

1. 2.

By symmetry, 1x

12 x31 xx

Or: 22 )3()1( xx 9612 22 xxxxx88 x 1

1 x

The Modulus Function

2

1y Ax x

B

x

2 xy y

Solutions:

2.

For B:

112 xxBy symmetry, at A:

3x

1312 xxSo,

12 x

13

( One region so one inequality )

(N.B. Strict inequality )

The Modulus Function

342 xxy

13

Method 2:

1442 xx

0342 xx

12 x

1)2( 2 x

Zeros: 0)1)(3( xx 13 xx or

12 xSo:

13 x

The Modulus Function

Activity

Card matchpair up the cards

to solve the inequalities

The Modulus Function

A x

x

5 xyB

x

22 xyy

2 5

Exercise

Solution: 1.

Solve the inequalities.

1. 522 xx 2.

For B,

)2(2)5( xx425 xx

x39 3x

3-1

For A, )2(2)5( xx or )2(2)5( xx 1x

32 xx

31522 xxx So,

The Modulus Function

322 xxy

Method 2: 522 xx

31522 xxxSo,

2510)44(4 22 xxxx

0)3)(1(3 xxZeros:

22 )5()2(4 xx

251016164 22 xxxx

0963 2 xx0)32(3 2 xx

31 xx or

The Modulus Function

3 xy

xy 2

y

x0

A

B

x

x

13

Exercise2. 32 xxSolutio

n:

For B,)3(2 xx

33 x

)3(2 xxFor A,

3x

1x

32 xxor:

1332 xxxx orSo, ( Two regions so two

inequalities )

The Modulus Function

0)3)(1(3 xxZeros:322 xxy

22 )3(4 xx

0)32(3 2 xx

31 xx or

Method 2: 32 xx

1332 xxxx orSo,

964 22 xxx0963 2 xx

13

The Modulus Function

The Modulus Function

The Modulus Function

The Modulus Function