ONE STEP EQUATIONS What you do to one side of the equation must also be done to the other side to...

ONE STEP EQUATIONS

ONE STEP EQUATIONS

What you do to one side of the What you do to one side of the equation must also be done to the equation must also be done to the other side to keep it balanced.other side to keep it balanced.

An equation is like a balance scale An equation is like a balance scale because it shows that two because it shows that two quantities are equal.quantities are equal.

ONE STEP EQUATIONS

• What is the variable?

To solve one step equations, you need to To solve one step equations, you need to ask three questions about the equation:ask three questions about the equation:

• What operation is performed on the variable?What operation is performed on the variable?

• What is the inverse operation? What is the inverse operation?

(The one that will undo what is being (The one that will undo what is being done to the variable)done to the variable)

ONE STEP EQUATIONSExample 1 Solve x + 4 = 12

What is the variable?What is the variable?

Using the Using the subtraction property of equalitysubtraction property of equality, subtract 4 from both sides , subtract 4 from both sides of the equation.of the equation.

- 4- 4

xx

x + 4 = 12x + 4 = 12

The variable is x.The variable is x.

Addition.Addition.

Subtraction.Subtraction.What is the inverse operation (the one that will undo what is being What is the inverse operation (the one that will undo what is being done to the variable)?done to the variable)?

What operation is being performed on the variable?What operation is being performed on the variable?

- 4- 4

88==

The subtraction property of equality tells us to subtract the The subtraction property of equality tells us to subtract the same thing on both sides to keep the equation equal.same thing on both sides to keep the equation equal.

ONE STEP EQUATIONSExample 2 Solve y - 7 = -13


Using the Using the addition property of equalityaddition property of equality, add 7 to both sides of the , add 7 to both sides of the equation.equation.

y - 7 = -13y - 7 = -13

+ 7+ 7

yy

The variable is y.The variable is y.

Subtraction.Subtraction.

Addition.Addition.

What operation is being performed on the variable?What operation is being performed on the variable?

What is the inverse operation (the one that will undo what is being done to What is the inverse operation (the one that will undo what is being done to the variable)? the variable)?

+ 7+ 7

== -6-6

The addition property of equality tells us to add the same thing on The addition property of equality tells us to add the same thing on both sides to keep the equation equal.both sides to keep the equation equal.

ONE STEP EQUATIONSExample 3 Solve –6a = 12


Using the Using the division property of equalitydivision property of equality, divide both sides of the , divide both sides of the equation by –6.equation by –6.

––6a = 126a = 12

-6-6

aa

The variable is a.The variable is a.

What operation is being performed on the variable?What operation is being performed on the variable? Multiplication.Multiplication.

What is the inverse operation (the one that will undo what is being done to the What is the inverse operation (the one that will undo what is being done to the variable)?variable)? DivisionDivision

-6-6

== -2-2

The division property of equality tells us to divide the same thing The division property of equality tells us to divide the same thing on both sides to keep the equation equal.on both sides to keep the equation equal.

ONE STEP EQUATIONSExample 4 Solve

2b


Using the Using the multiplication property of equalitymultiplication property of equality, multiply both sides of , multiply both sides of the equation by 2.the equation by 2.

= -10= -102b

2b

bb

The variable is b.The variable is b.What operation is being performed on the variable?What operation is being performed on the variable? Division. Division. What is the inverse operation (the one that will undo what is being done What is the inverse operation (the one that will undo what is being done to the variable)?to the variable)? MultiplicationMultiplication

= -10= -102 2 •• • • 22

== -20-20

== -10-10

The multiplication property of equality tells us to multiply the The multiplication property of equality tells us to multiply the same thing on both sides to keep the equation equal.same thing on both sides to keep the equation equal.

SolvingSolving Two-Step Two-Step EquationsEquations

What is a Two-Step Equation?

An equation written in the form

Ax + B = C

Examples of Two-Step Equations

a) 3x – 5 = 16

b) y/4 + 3 = 12

c) 5n + 4 = 6

d) n/2 – 6 = 4

Steps for Solving Two-Step Equations

1. Solve for any Addition or Subtraction on the variable side of equation by “undoing” the operation from both sides of the equation.

2. Solve any Multiplication or Division from variable side of equation by “undoing” the operation from both sides of the equation.

Opposite Operations

Addition Subtraction

Multiplication Division

Helpful Hints?

Identify what operations are on the variable side. (Add, Sub, Mult, Div)

“Undo” the operation by using opposite operations.

Whatever you do to one side, you must do to the other side to keep equation balanced.

Ex. 1: Solve 4x – 5 = 11

4x – 5 = 15 +5 +5 (Add 5 to both sides)4x = 20 (Simplify) 4 4 (Divide both sides

by 4)x = 5 (Simplify)

Try These Examples1. 2x – 5 = 17

2. 3y + 7 = 25

3. 5n – 2 = 38

4. 12b + 4 = 28

Check your answers!!!1. x = 11

2. y = 6

3. n = 8

4. b = 2

Ready to Move on?

Ex. 2: Solve x/3 + 4 = 9

x/3 + 4 = 9 - 4 - 4 (Subt. 4 from both sides)x/3 = 5 (Simplify)

(x/3) 3 = 5 3 (Mult. by 3 on both sides)

x = 15 (Simplify)

Try these examples!

1. x/5 – 3 = 8

2. c/7 + 4 = 9

3. r/3 – 6 = 2

4. d/9 + 4 = 5

Check your answers!!!

1. x = 55

2. c = 35

3. r = 24

4. d = 9

Time to Review!

• Make sure your equation is in the form Ax + B = C

• Keep the equation balanced.• Use opposite operations to “undo”

• Follow the rules:1. Undo Addition or Subtaction2. Undo Multiplication or Division

Before we begin…

• In previous lessons we solved simple one-step linear equations…

• In this lesson we will look at solving multi-step equations…as the name suggests there are steps you must do before you can solve the equation…

• To be successful here you have to be able to analyze the equation and decide what steps must be taken to solve the equation…

Multi-Step Equations

• This lesson will look at:– Solving linear equations with 2 operations

– Combining like terms first

– Using the distributive property

– Distributing a negative

– Multiplying by a reciprocal first

• Your ability to be organized and lay out your problem will enable you to be successful with these concepts

Linear Equations with 2 Functions

• Sometimes linear equations will have more than one operation

• In this instance operations are defined as add, subtract, multiply or divide

• The rule is – First you add or subtract – Second you multiply or divide.

Linear Equations with 2 Operations

• You will have to analyze the equation first to see what it is saying…

• Then, based upon the rules, undo each of the operations…

• The best way to explain this is by looking at an example….

Example # 1

2x + 6 = 162x + 6 = 16

In this case the problem says 2 times a number plus 6 is equal to In this case the problem says 2 times a number plus 6 is equal to 1616 TimesTimes means to means to multiplymultiply and and plusplus means to means to addadd

The first step is to add or subtractThe first step is to add or subtract..

To undo the addition of 6 I have to subtract 6 from both sides To undo the addition of 6 I have to subtract 6 from both sides which looks like this:which looks like this:

2x + 6 = 162x + 6 = 16 - 6 -6- 6 -6

2x = 102x = 10

The 6’s The 6’s on the left on the left

side side cancel cancel

each other each other outout

16 minus 16 minus 6 equals 6 equals

1010

What is left is a one step What is left is a one step equation 2x = 10equation 2x = 10

Example #1 (Continued) 2x = 102x = 10

To undo the multiplication here you would divide both sides To undo the multiplication here you would divide both sides by 2 which looks like this:by 2 which looks like this:

2x = 102x = 10

2 22 2The 2’s on The 2’s on

the left the left cancel out cancel out leaving xleaving x

x = 5x = 5

10 divided 10 divided by 2 is by 2 is

equal to 5equal to 5

The solution to the equation 2x + 6 = 16 is The solution to the equation 2x + 6 = 16 is x = 5x = 5

The second step is to multiply or divideThe second step is to multiply or divide

Combining Like Terms First• When solving multi-step equations, sometimes you have

to combine like terms first.

• The rule for combining like terms is that the terms must have the same variable and the same exponent.

Example:

• You can combine x + 5x to get 6x

• You cannot combine x + 2x2 because the terms do not have the same exponent

Example # 27x – 3x – 8 = 247x – 3x – 8 = 24

I begin working on the left side of the equationI begin working on the left side of the equation

On the left side I notice that I have two like terms (7x, -3x) On the left side I notice that I have two like terms (7x, -3x) since the terms are alike I can combine them to get 4x.since the terms are alike I can combine them to get 4x.

7x – 3x – 8 = 247x – 3x – 8 = 24

4x – 8 = 244x – 8 = 24

After I combine the terms I have a 2-step equation. To solve After I combine the terms I have a 2-step equation. To solve this equation add/subtract 1this equation add/subtract 1stst and then multiply/divide and then multiply/divide

Example # 2 (continued)4x – 8 = 244x – 8 = 24

Step 1: Add/SubtractStep 1: Add/Subtract

Since this equation has – 8, I will add 8 to both Since this equation has – 8, I will add 8 to both sidessides 4x – 8 = 244x – 8 = 24

+8 +8+8 +8

4x = 324x = 32The 8’s The 8’s on the on the

left left cancel cancel

outout

24 + 8 = 3224 + 8 = 32

I am left with a 1-step I am left with a 1-step equationequation

Example # 2 (continued)

4x = 324x = 32

Step 2: Multiply/DivideStep 2: Multiply/Divide

In this instance 4x means 4 times x. To undo the In this instance 4x means 4 times x. To undo the multiplication divide both sides by 4multiplication divide both sides by 4

4x = 324x = 32

4 44 4The 4’s on The 4’s on the left the left

cancel out cancel out leaving xleaving x

x = 8x = 8

32 32 4 = 8 4 = 8

The solution that makes the The solution that makes the statement true is x = 8statement true is x = 8

Solving equations using the Distributive Property

• When solving equations, sometimes you will need to use the distributive property first.

• At this level you are required to be able to recognize and know how to use the distributive property

• Essentially, you multiply what’s on the outside of the parenthesis with EACH term on the inside of the parenthesis

• Let’s see what that looks like…

Example #35x + 3(x +4) = 285x + 3(x +4) = 28

In this instance I begin on the left side of the equationIn this instance I begin on the left side of the equation

I recognize the distributive property as 3(x +4). I must simplify I recognize the distributive property as 3(x +4). I must simplify that before I can do anything elsethat before I can do anything else

5x + 3(x +4) = 285x + 3(x +4) = 28

+3x+3x +12+125x5x = 28= 28After I do the distributive property I see that I have like terms After I do the distributive property I see that I have like terms (5x and 3x) I have to combine them to get 8x before I can (5x and 3x) I have to combine them to get 8x before I can solve this equationsolve this equation

Example # 3(continued)8x + 12 = 288x + 12 = 28

I am now left with a 2-step equationI am now left with a 2-step equation

Step 1: Add/SubtractStep 1: Add/Subtract

The left side has +12. To undo the +12, I subtract 12 from both The left side has +12. To undo the +12, I subtract 12 from both sidessides 8x + 12 = 288x + 12 = 28

-12 -12-12 -12

8x = 168x = 16

The 12’s The 12’s on the left on the left cancel out cancel out leaving 8xleaving 8x

28 – 12 = 1628 – 12 = 16


8x = 168x = 16

Step 2: Multiply/DivideStep 2: Multiply/Divide

On the left side 8x means 8 times x. To undo the multiplication On the left side 8x means 8 times x. To undo the multiplication I divide both sides by 8I divide both sides by 8

8x = 168x = 16

8 88 8The 8’s on The 8’s on the left the left cancel out cancel out leaving xleaving x

16 16 8 = 2 8 = 2

The solution that makes the statement true is x = 2The solution that makes the statement true is x = 2

x = 2x = 2

Distributing a Negative• Distributing a negative number is similar to

using the distributive property.• However, students get this wrong because

they forget to use the rules of integers• Quickly the rules are…when multiplying, if

the signs are the same the answer is positive. If the signs are different the answer is negative

Example #4

4x – 3(x – 2) = 214x – 3(x – 2) = 21

I begin by working on the left side of the equation.I begin by working on the left side of the equation.

In this problem I have to use the distributive property. In this problem I have to use the distributive property. However, the 3 in front of the parenthesis is a negative 3.However, the 3 in front of the parenthesis is a negative 3.

When multiplying here, multiply the -3 by both terms within When multiplying here, multiply the -3 by both terms within the parenthesis. Use the rules of integersthe parenthesis. Use the rules of integers

4x – 3(x – 2) = 214x – 3(x – 2) = 21

4x4x – – 3x3x + + 66 = 21= 21

After doing the distributive property, I see that I can combine After doing the distributive property, I see that I can combine the 4x and the -3x to get 1x or xthe 4x and the -3x to get 1x or x


4x4x – – 3x3x + + 66 = 21= 21

x = 15x = 15

After combining like terms you are left with a simple one step After combining like terms you are left with a simple one step equation. To undo the +6 subtract 6 from both sides of the equationequation. To undo the +6 subtract 6 from both sides of the equation

x + 6 = 21x + 6 = 21

-6 -6-6 -6

x + 6 = 21x + 6 = 21

The 6’s The 6’s cancel cancel

out out leaving leaving

xx

21 – 6 = 1521 – 6 = 15

The solution is x = 15The solution is x = 15

Multiplying by a Reciprocal First• Sometimes when doing the distributive property

involving fractions you can multiply by the reciprocal first.

• Recall that the reciprocal is the inverse of the fraction and when multiplied their product is equal to 1.

• The thing about using the reciprocal is that you have to multiply both sides of the equation by the reciprocal.

• Let’s see what that looks like…

Example # 512

3

102 ( )x

In this example you could distribute the 3/10 to the x and the In this example you could distribute the 3/10 to the x and the 2.2.The quicker way to handle this is to use the reciprocal of 3/10 The quicker way to handle this is to use the reciprocal of 3/10 which is 10/3 and multiply both sides of the equation by 10/3which is 10/3 and multiply both sides of the equation by 10/3

1210

3

10

3

3

102F

HGIKJFHG

IKJ ( )x

On the On the left sideleft side of the equation, after multiplying by the of the equation, after multiplying by the reciprocal 10/3 you are left with 120/3 which can be simplified reciprocal 10/3 you are left with 120/3 which can be simplified to 40 to 40 On the On the right sideright side of the equation the reciprocals cancel each of the equation the reciprocals cancel each other out leaving x + 2other out leaving x + 2

The new equation is:The new equation is:

40 = x + 240 = x + 2

Example #5 (continued)

38 = x38 = x

After using the reciprocals you are left with a simple one-step After using the reciprocals you are left with a simple one-step equationequationTo solve this equation begin by working on the right side To solve this equation begin by working on the right side and subtract 2 from both sides of the equationand subtract 2 from both sides of the equation

40 = x + 240 = x + 2

- 2 -2- 2 -2

40 = x + 240 = x + 2

40 – 2 = 3840 – 2 = 38The 2’s on The 2’s on the right the right

side side cancel out cancel out leaving xleaving x

The solution to the equation is x = 38The solution to the equation is x = 38

Comments• On the next couple of slides are some practice

problems…The answers are on the last slide…

• Do the practice and then check your answers…If you do not get the same answer you must question what you did…go back and problem solve to find the error…

• If you cannot find the error bring your work to me and I will help…

Your Turn

1. 6x – 4(9 –x) = 106

2. 2x + 7 = 15

3. 6 = 14 – 2x

4. 3(x – 2) = 18

5. 12(2 – x) = 6

Your Turn9

23 27x b g

4

92 4 48xb g

6.6.

7.7.

8.8.

9.9.

10.10.

5m – (4m – 1) = -125m – (4m – 1) = -12

55x – 3(9x + 12) = -6455x – 3(9x + 12) = -64

9x – 5(3x – 12) = 309x – 5(3x – 12) = 30

Your Turn Solutions

1. 14.2

2. 4

3. 4

4. 8

5. 1 ½

6. 3

7. -52

8. -13

9. -1

10. 5

Warm UpWarm UpSolve each equation. Solve each equation.

1. 1. 22xx – 5 = –17 – 5 = –17

2. 2.

Solve each inequality and graph the Solve each inequality and graph the solutions.solutions.

4. 4.

3. 3. 5 < 5 < tt + 9 + 9

––66

1414

tt > –4 > –4

aa ≤ –8 ≤ –8

Solving Multi-Step InequalitiesSolving Multi-Step Inequalities

Solve the inequality and graph the solutions.Solve the inequality and graph the solutions.

45 + 245 + 2bb > 61 > 61

45 + 245 + 2bb > 61 > 61––45 –4545 –45

22b > b > 1616

bb > 8 > 8

00 22 44 66 88 1010 1212 1414 1616 1818 2020

1. Add/ Subtract1. Add/ Subtract

2. Multiply/Divide2. Multiply/Divide

The solution set is {b:b > 8}.The solution set is {b:b > 8}.

8 – 38 – 3yy ≥ 29 ≥ 29

8 – 38 – 3yy ≥ 29 ≥ 29

––8 –88 –8

––33y y ≥≥ 2121

yy ≤≤ –7 –7

1. Add/Subtract1. Add/Subtract


––1010 ––88 ––66 ––44 ––22 00 22 44 66 88 1010

––77

Solving Multi-Step InequalitiesSolving Multi-Step Inequalities


The solution set is {y:y The solution set is {y:y –7}. –7}.

You try

Solve and Graph the inequality

To solve more complicated inequalities, you may To solve more complicated inequalities, you may first need to simplify the expressions on one or first need to simplify the expressions on one or both sides.both sides.

Simplifying Before Solving InequalitiesSimplifying Before Solving Inequalities


2 – (–10) > –42 – (–10) > –4tt

12 > –412 > –4tt

––3 < 3 < tt (or (or t > t > –3) –3)

1. Combine like terms.1. Combine like terms.


––33

––1010 ––88 ––66 ––44 ––22 00 22 44 66 88 1010

The solution set is {t:t > –3}. The solution set is {t:t > –3}.

Simplifying Before Solving InequalitiesSimplifying Before Solving InequalitiesSolve the inequality and graph the solutions.Solve the inequality and graph the solutions.

––4(2 – 4(2 – xx) ≤ 8) ≤ 8

−−44(2 – (2 – xx) ≤ 8) ≤ 8

−−44(2) (2) − 4− 4(−(−xx) ≤ 8) ≤ 8

––8 + 48 + 4xx ≤ 8 ≤ 8+8 +8+8 +8

44x x ≤≤ 1616

x x ≤ 4≤ 4

1. Distributive Property1. Distributive Property

2. Add/Subtract2. Add/Subtract


––1010 ––88 ––66 ––44 ––22 00 22 44 66 88 1010

The solution set is {x:x ≤ 4}. The solution set is {x:x ≤ 4}.

Now you try…Now you try…

x > 3x > 3

x ≥ -1x ≥ -1

x ≤ 2x ≤ 2

x < x < 3535

x ≥ -x ≥ -2727

1. 3x – 7 > 21. 3x – 7 > 2 4. 4. xx – 4 < 3 – 4 < 3

55

2. 4x + 1 2. 4x + 1 -3 -3

5. 15 + 5. 15 + xx ≥ 6 ≥ 6

33

3. 2x – 7 ≤ -33. 2x – 7 ≤ -3

Lesson Quiz: Part ILesson Quiz: Part ISolve each inequality and graph the solutions.Solve each inequality and graph the solutions.

1. 1. 13 – 213 – 2xx ≥ 21 ≥ 21 x x ≤≤ –4 –4

2. 2. –11 + 2–11 + 2 < < 33pp p p >> –3–3

3. 3. 2233 < –2(3 – < –2(3 – tt)) tt > 7 > 7

4.4.

Copyright © 2010 Pearson Education, Inc. All rights reserved.

Sec 3.3 - 56

3.3 Absolute Value Equations and InequalitiesSummary:

Solving Absolute Value Equations and Inequalities

1. 1. To solve To solve ||ax ax + + bb| = | = kk, , solve the following compound equation.solve the following compound equation.

Let Let kk be a positive real number, and be a positive real number, and pp and and qq be real numbers. be real numbers.

ax ax + + bb = = kk oror ax ax + + bb = – = –k.k.

The solution set is usually of the form {The solution set is usually of the form {pp, , qq}, which includes two}, which includes two

numbers.numbers.

pp qq


Sec 3.3 - 57



2. 2. To solve To solve ||ax ax + + bb| > | > kk, , solve the following compound inequality.solve the following compound inequality.


ax ax + + bb > > kk oror ax ax + + bb < – < –k.k.

The solution set is of the form (-The solution set is of the form (-∞∞, , pp) U () U (qq, , ∞∞), which consists of two), which consists of two

separate intervals. separate intervals.

pp qq


Sec 3.3 - 58



3. 3. To solve To solve ||ax ax + + bb| < | < kk, , solve the three-part inequalitysolve the three-part inequality


––k k << ax ax + + bb < < kk

The solution set is of the form (The solution set is of the form (pp,, q q), a single interval.), a single interval.

pp qq

3.3 Absolute Value Equations and Inequalities

EXAMPLE 1 Solving an Absolute Value EquationSolving an Absolute Value Equation

Solve |2Solve |2xx + 3| = 5. + 3| = 5.


EXAMPLE 2 Solving an Absolute Value Inequality with >Solving an Absolute Value Inequality with >

Solve |2Solve |2xx + 3| > 5. + 3| > 5.


EXAMPLE 3 Solving an Absolute Value Inequality with <Solving an Absolute Value Inequality with <

Solve |2Solve |2xx + 3| < 5. + 3| < 5.

3.3 Absolute Value Equations and InequalitiesEXAMPLE 4 Solving an Absolute Value Equation That Solving an Absolute Value Equation That

Requires RewritingRequires Rewriting

Solve the equation |Solve the equation |xx – 7| + 6 = 9. – 7| + 6 = 9.


Sec 3.3 - 63


Special Cases for Absolute Value

Special Cases for Absolute ValueSpecial Cases for Absolute Value

1.1. The absolute value of an expression can never be negative: |The absolute value of an expression can never be negative: |aa| ≥ 0| ≥ 0

for all real numbers for all real numbers aa..

2.2. The absolute value of an expression equals 0 only when the The absolute value of an expression equals 0 only when the expression is equal to 0. expression is equal to 0.


Sec 3.3 - 64

3.3 Absolute Value Equations and InequalitiesEXAMPLE 6 Solving Special Cases of Absolute ValueSolving Special Cases of Absolute Value

EquationsEquations

Solve each equation.Solve each equation.

See Case 1 in the preceding slide. Since the absolute value of an See Case 1 in the preceding slide. Since the absolute value of an expression can never be negative, there are no solutions for this equation.expression can never be negative, there are no solutions for this equation.The solution set is Ø.The solution set is Ø.

(a)(a) |2|2nn + 3| = + 3| = –7–7

See Case 2 in the preceding slide. The absolute value of the expres-See Case 2 in the preceding slide. The absolute value of the expres-sion 6sion 6ww – 1 will equal 0 – 1 will equal 0 only only ifif

66ww – 1 = 0. – 1 = 0.

(b)(b) |6|6ww – 1| = – 1| = 00

The solution of this equation is . Thus, the solution set of the originalThe solution of this equation is . Thus, the solution set of the original

equation is { }, with just one element. Check by substitution.equation is { }, with just one element. Check by substitution.

1166

1166


Sec 3.3 - 65


InequalitiesInequalities

Solve each inequality.Solve each inequality.

The absolute value of a number is always greater than or equal to 0.The absolute value of a number is always greater than or equal to 0.Thus, |Thus, |xx| ≥| ≥ –2 is true for –2 is true for all all real numbers. The solution set is (–real numbers. The solution set is (–∞∞, , ∞∞). ).

(a)(a) ||xx| ≥| ≥ –2–2

Add 1 to each side to get the absolute value expression alone on oneAdd 1 to each side to get the absolute value expression alone on oneside.side.

||xx + 5| < –7 + 5| < –7

(b)(b) ||xx + 5| – 1 < –8 + 5| – 1 < –8

There is no number whose absolute value is less than –7, so this inequalityThere is no number whose absolute value is less than –7, so this inequalityhas no solution. The solution set is Ø.has no solution. The solution set is Ø.


Sec 3.3 - 66


InequalitiesInequalities

Solve each inequality.Solve each inequality.

Subtracting 2 from each side givesSubtracting 2 from each side gives

||xx – 9| ≤ 0 – 9| ≤ 0

(c)(c) ||xx – 9| + 2 ≤ 2 – 9| + 2 ≤ 2

The value of |The value of |xx – 9| will never be less than 0. However, | – 9| will never be less than 0. However, |xx – 9| will equal 0 – 9| will equal 0when when xx = 9. Therefore, the solution set is {9}. = 9. Therefore, the solution set is {9}.

This can be written as This can be written as 1 1 x x 7 7..

SolveSolve | | xx 4 4 | < 3| < 3

Solving an Absolute-Value InequalitySolving an Absolute-Value Inequality

xx 4 4 IS POSITIVEIS POSITIVE xx 4 4 IS NEGATIVEIS NEGATIVE

| | xx 4 4 | | 3 3

xx 4 4 33

xx 7 7

| | xx 4 4 | | 3 3

xx 4 4 33

xx 1 1

Reverse inequality Reverse inequality symbolsymbol..

The solution is all real numbers greater than 1 The solution is all real numbers greater than 1 andand less than 7 less than 7..

22xx 1 1 99

|| 2 2xx 1 1 || 3 3 6 6

|| 2 2xx 1 1 || 9 9

22xx 1010

22xx + 1 + 1 IS NEGATIVEIS NEGATIVE

xx 55

Solve Solve | 2| 2xx 1 1 | | 3 3 6 6 and graph the solution.and graph the solution.

|| 2 2xx 1 1 || 3 3 6 6

|| 2 2xx 1 1 || 9 9

22xx 1 1 +9 +9

22xx 8 8

22xx + 1 + 1 IS POSITIVEIS POSITIVE

xx 4 4

Solving an Absolute-Value InequalitySolving an Absolute-Value Inequality

Reverse Reverse inequality inequality symbol.symbol.

|| 2 2xx 1 1 || 3 3 6 6

|| 2 2xx 1 1 || 9 9

22xx 1 1 +9 +9

xx 4 422xx 8 8

|| 2 2xx 1 1 || 3 3 6 6

|| 2 2xx 1 1 || 9 9

22xx 1 1 9922xx 1010xx 55

22xx + 1 + 1 IS POSITIVEIS POSITIVE 22xx + 1 + 1 IS NEGATIVEIS NEGATIVE

6 6 5 5 4 4 3 3 2 2 1 0 1 2 3 4 5 6 1 0 1 2 3 4 5 6

The solution is all real numbers greater than or equal The solution is all real numbers greater than or equal to to 44 oror less than or equal to less than or equal to 55. This can be written as . This can be written as the compound inequality the compound inequality x x 55 oror xx 4 4.. 55 44..

Strange Results

57)83(2 x True for True for All Real NumbersAll Real Numbers, , since absolute value is since absolute value is alwaysalways positive, and positive, and therefore greater than any therefore greater than any negative.negative.

2)12)]48(3[2( 3 x No Solution No Solution ØØ. . Positive numbers are Positive numbers are nevernever less than less than negative numbers. negative numbers.

Examples752 w

752 w 752 woror

122 w6w

22 w1woror

Check and verify on a number line. Numbers above 6 or below Check and verify on a number line. Numbers above 6 or below -1 keep the absolute value greater than 7. Numbers between -1 keep the absolute value greater than 7. Numbers between

them make the absolute value less than 7. them make the absolute value less than 7.

Key Skills

Solve absolute-value inequalities.Solve absolute-value inequalities.

Solve Solve ||xx – 4 – 4| | 5. 5.

xx – 4 is positive – 4 is positive

xx – 4 – 4 5 5

xx – 4 is negative – 4 is negative

xx 9 9

Case 1:Case 1: Case 2:Case 2:

xx – 4 – 4 –5 –5

xx –1 –1

solution: –1 solution: –1 xx 9 9

Key Skills


Solve Solve |4|4xx – 2 – 2|| -18. -18. Exception Exception alert!!!!alert!!!!

When the absolute value When the absolute value equals a negative value, equals a negative value, there is no solution. there is no solution.

Key Skills


Solve Solve |2|2xx – 6 – 6|| 18. 18.

2x2x – 6 is positive – 6 is positive2x2x – 6 – 6 18 18

2x2x – 6 is negative – 6 is negative

xx 12 12

Case 1:Case 1: Case 2:Case 2:

2x - 6 2x - 6 –18 –18

xx –6 –6Solution: –6 Solution: –6 xx 12 12

2x2x 24 24 2x 2x –12 –12

TRY THISTRY THIS

Key Skills


Solve Solve |3|3xx – 2 – 2|| -4. -4. Exception Exception alert!!!!alert!!!!

When the absolute value When the absolute value equals a negative value, equals a negative value, there is no solution. there is no solution.

TRY THISTRY THIS

ONE STEP EQUATIONS What you do to one side of the equation must also be done to the other side to...

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Transcript of ONE STEP EQUATIONS What you do to one side of the equation must also be done to the other side to...