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Equations with Absolute Value. Equations with Absolute Value Essential Question. M8A1c. Solve algebraic equations or inequalities in one variable, including those involving absolute values. How does absolute value affect the solution of an equation?. 4 units 4 units.

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Equations with absolute value

Equations with

Absolute Value


Equations with absolute value

Equations with Absolute Value Essential Question

M8A1c. Solve algebraic equations or inequalities in one variable, including those involving absolute values.

How does absolute value affect the solution of an equation?


Equations with absolute value

4 units 4 units

–5 –4 –3 –2 –1 0 1 2 3 4 5

A number’s absolute value is its distance from 0 on a number line. Absolute value is always positive because distance is always positive. β€œThe absolute value of –4” is written as |–4|. Opposites have the same absolute value.

|–4| = 4

|4| = 4


Equations with absolute value

3 units

–5 –4 –3 –2 –1 0 1 2 3 4 5

Additional Example 3: Simplifying Absolute-Value Expressions

Simplify each expression.

A. |–3|

–3 is 3 units from 0, so |–3| = 3.

B. |17 – 6|

|17 – 6| = |11|

Subtract first: 17 – 6 = 11.

Then find the absolute value: 11 is 11 units from 0.

= 11


Equations with absolute value

Additional Example 3: Simplifying Absolute-Value Expressions

Simplify each expression.

C. |–8| + |–5|

Find the absolute values first: –8 is 8 units from 0. –5 is 5 units from 0. Then add.

|–8| + |–5| = 8 + 5

= 13

D. |5 + 1| + |8 – 6|

|5 + 1| + |8 – 6| = |6| + |2|

5 + 1 = 6, 8 – 6 = 2.

6 is 6 units from 0, 2 is 2 units from 0. Add.

= 6 + 2

= 8


Equations with absolute value

5 units

–5 –4 –3 –2 –1 0 1 2 3 4 5

Check It Out! Example 3

Simplify each expression.

A. |–5|

–5 is 5 units from 0, so |–5| = 5.

B. |12 – 4|

|12 – 4| = |8|

Subtract first: 12 – 4 = 8.

Then find the absolute value: 8 is 8 units from 0.

= 8


Equations with absolute value

Check It Out! Example 3

Simplify each expression.

C. |–2| + |–9|

Find the absolute values first: –2 is 2 units from 0. –9 is 9 units from 0. Then add.

|–2| + |–9| = 2 + 9

= 11

D. |3 + 1| + |9 – 2|

|3 + 1| + |9 – 2| = |4| + |7|

3 + 1 = 4, 9 – 2 = 7.

4 is 4 units from 0, 7 is 7 units from 0. Add.

= 4 + 7

= 11


Absolute value of x

Symbol β”‚xβ”‚

The distance x is from 0 on the number line.

Always positive

Ex: β”‚-3β”‚=3

Absolute Value (of x)

-4 -3 -2 -1 0 1 2


Equations with absolute value

You can solve some absolute-value equations using mental math. For instance, you learned that the equation |x|ο€½ 8 has two solutions: 8 and ο€­8.

To solve absolute-value equations, you can use the fact that the expression inside the absolute value symbols can be either positive or negative.


Equations with absolute value

First, isolate the absolute-value expression on the left side of the equality.

β”‚ ax + b β”‚= c

Rewrite as Positive Case

ax + b = c

and solve for x.

Rewrite as Negative case

ax + b = -c

and solve for x.

Solve 6x – 3 = 15

Solve x – 2 = 5

Graphic Organizer by Dale Graham and Linda Meyer

Thomas County Central High School; Thomasville GA


Equations with absolute value

Solving an Absolute-Value Equation side of the equality.

The expressionx ο€­ 2 can be equal to 5 or ο€­5.

x ο€­ 2IS NEGATIVE

| x ο€­ 2 | ο€½ 5

x ο€­ 2IS POSITIVE

x ο€­ 2 IS POSITIVE

x ο€­ 2 IS POSITIVE

x ο€­ 2 IS NEGATIVE

x ο€­ 2 ο€½ ο€­5

| x ο€­ 2 | ο€½ 5

| x ο€­ 2 | ο€½ 5

x ο€½ ο€­3

x ο€­ 2 ο€½ 5

x ο€­ 2 ο€½ 5

x ο€­ 2 ο€½ 5

x ο€­ 2 ο€½ ο€­5

CHECK

x ο€½ 7

x ο€½ 7

Solve | x ο€­ 2 | ο€½ 5

Solve | x ο€­ 2 | ο€½ 5

SOLUTION

The expressionx ο€­ 2can be equal to5orο€­5.

x ο€­ 2 IS POSITIVE

x ο€­ 2 IS NEGATIVE

| x ο€­ 2 | ο€½ 5

| x ο€­ 2 | ο€½ 5

x ο€­ 2 ο€½ 5

x ο€­ 2 ο€½ ο€­5

x ο€½ 7

x ο€½ ο€­3

The equation has two solutions: 7 and –3.

| 7 ο€­ 2 | ο€½ | 5 | ο€½ 5

|ο€­3 ο€­ 2 | ο€½ | ο€­5 | ο€½ 5


Ex solve 6x 3 15

6x-3 = 15 or 6x-3 = -15 side of the equality.

6x = 18 or 6x = -12

x = 3 or x = -2

* Plug in answers to check your solutions!

Ex: Solve 6x-3 = 15


Equations with absolute value

Solving an Absolute-Value Equation side of the equality.

Isolate theabsolute value expressionon one side of the equation.

2x ο€­ 7 IS NEGATIVE

2x ο€­ 7 IS POSITIVE

2x ο€­ 7 IS NEGATIVE

2x ο€­ 7 IS POSITIVE

2x ο€­ 7 IS POSITIVE

| 2x ο€­ 7 | ο€­ 5 ο€½ 4

| 2x ο€­ 7 | ο€­ 5 ο€½ 4

| 2x ο€­ 7 | ο€­ 5 ο€½ 4

| 2x ο€­ 7 | ο€½ 9

| 2x ο€­ 7 | ο€½ 9

| 2x ο€­ 7 | ο€½ 9

2x ο€­ 7 ο€½ ο€­9

2x ο€­ 7 ο€½ +9

2x ο€­ 7 ο€½ ο€­9

2x ο€­ 7 ο€½ +9

2x ο€­ 7 ο€½ +9

2x ο€½ ο€­2

2x ο€½ 16

2x ο€½ 16

x ο€½ 8

x ο€½ ο€­1

x ο€½ ο€­1

TWO SOLUTIONS

x ο€½ 8

x ο€½ 8

Solve | 2x ο€­ 7 | ο€­ 5 ο€½ 4

Solve | 2x ο€­ 7 | ο€­ 5 ο€½ 4

SOLUTION

Isolate theabsolute value expressionon one side of the equation.

2x ο€­ 7 IS POSITIVE

2x ο€­ 7 IS NEGATIVE

| 2x ο€­ 7 | ο€­ 5 ο€½ 4

| 2x ο€­ 7 | ο€­ 5 ο€½ 4

| 2x ο€­ 7 | ο€½ 9

| 2x ο€­ 7 | ο€½ 9

2x ο€­ 7 ο€½ +9

2x ο€­ 7 ο€½ ο€­9

2x ο€½ 16

2x ο€½ ο€­2

x ο€½ 8

x ο€½ ο€­1


Ex solve 2x 7 3 8

Get the absolute value part by itself first! side of the equality.

2x+7 = 11

Now split into 2 parts.

2x+7 = 11 or 2x+7 = -11

2x = 4 or 2x = -18

x = 2 or x = -9

Check the solutions.

Ex: Solve 2x + 7 -3 = 8