Absolute Value Equations

The geometric meaning of the absolute value of x, denoted as |x|, is the distance measured from x to 0 on the real line.

Absolute Value Equations

0 5

Hence | 5 | = 5

The geometric meaning of the absolute value of x, denoted as |x|, is the distance measured from x to 0 on the real line.

a distance of 5

Absolute Value Equations

0 5

Hence | 5 | = 5

| 5 | = 5

The geometric meaning of the absolute value of x, denoted as |x|, is the distance measured from x to 0 on the real line.

a distance of 5

Absolute Value Equations

The geometric meaning of the absolute value of x, denoted as |x|, is the distance measured from x to 0 on the real line.

0 5-5

Hence | 5 | = 5 = | -5 |

| 5 | = 5

a distance of 5

Absolute Value Equations

a distance of 5

0 5-5

Hence | 5 | = 5 = | -5 | = distance from 5 or –5 to 0.

| -5 | = 5 | 5 | = 5

The geometric meaning of the absolute value of x, denoted as |x|, is the distance measured from x to 0 on the real line.

a distance of 5a distance of 5

Absolute Value Equations

0 5-5

Hence | 5 | = 5 = | -5 | = distance from 5 or –5 to 0.

| -5 | = 5 | 5 | = 5

Because it is a distance measurement, |x| is nonnegative.

The geometric meaning of the absolute value of x, denoted as |x|, is the distance measured from x to 0 on the real line.

Absolute Value Equations

0 5-5

Hence | 5 | = 5 = | -5 | = distance from 5 or –5 to 0.

| -5 | = 5 | 5 | = 5

Because it is a distance measurement, |x| is nonnegative. Algebraic definition of absolute value

The geometric meaning of the absolute value of x, denoted as |x|, is the distance measured from x to 0 on the real line.

Absolute Value Equations

|x|= x if x is positive or 0. {

0 5-5

Hence | 5 | = 5 = | -5 | = distance from 5 or –5 to 0.

| -5 | = 5 | 5 | = 5

Because it is a distance measurement, |x| is nonnegative. Algebraic definition of absolute value

The geometric meaning of the absolute value of x, denoted as |x|, is the distance measured from x to 0 on the real line.

Absolute Value Equations

|x|= x if x is positive or 0. –x (opposite of x) if x is negative.{

0 5-5

Hence | 5 | = 5 = | -5 | = distance from 5 or –5 to 0.

| -5 | = 5 | 5 | = 5

Because it is a distance measurement, |x| is nonnegative. Algebraic definition of absolute value

The geometric meaning of the absolute value of x, denoted as |x|, is the distance measured from x to 0 on the real line.

Absolute Value Equations

|x|= x if x is positive or 0. –x (opposite of x) if x is negative.{

Hence | –5 | =

0 5-5

Hence | 5 | = 5 = | -5 | = distance from 5 or –5 to 0.

| -5 | = 5 | 5 | = 5

Because it is a distance measurement, |x| is nonnegative. Algebraic definition of absolute value

The geometric meaning of the absolute value of x, denoted as |x|, is the distance measured from x to 0 on the real line.

Absolute Value Equations

|x|= x if x is positive or 0. –x (opposite of x) if x is negative.{

Hence | –5 | = –(–5)

0 5-5

Hence | 5 | = 5 = | -5 | = distance from 5 or –5 to 0.

| -5 | = 5 | 5 | = 5

Because it is a distance measurement, |x| is nonnegative. Algebraic definition of absolute value

The geometric meaning of the absolute value of x, denoted as |x|, is the distance measured from x to 0 on the real line.

Absolute Value Equations

Because –5 is negative, use the 2nd part of the rule and take its opposite to obtainits abs. value.

|x|= x if x is positive or 0. –x (opposite of x) if x is negative.{

Hence | –5 | = –(–5) = 5.

0 5-5

Hence | 5 | = 5 = | -5 | = distance from 5 or –5 to 0.

| -5 | = 5 | 5 | = 5

Because it is a distance measurement, |x| is nonnegative. Algebraic definition of absolute value

The geometric meaning of the absolute value of x, denoted as |x|, is the distance measured from x to 0 on the real line.

Absolute Value Equations

Because –5 is negative, use the 2nd part of the rule and take its opposite to obtain its abs. value.

|x|= x if x is positive or 0. –x (opposite of x) if x is negative.{

Hence | –5 | = –(–5) = 5. Since the absolute value is never negative, an equation such as |x14 – 3x9 + 1 | = –2 doesn't have any solution.

0 5-5

Hence | 5 | = 5 = | -5 | = distance from 5 or –5 to 0.

| -5 | = 5 | 5 | = 5

Because it is a distance measurement, |x| is nonnegative. Algebraic definition of absolute value

The geometric meaning of the absolute value of x, denoted as |x|, is the distance measured from x to 0 on the real line.

Absolute Value Equations

Because –5 is negative, use the 2nd part of the rule and take its opposite to obtain its abs. value.

|x|= x if x is positive or 0. –x (opposite of x) if x is negative.{

Hence | –5 | = –(–5) = 5. Since the absolute value is never negative, an equation such as |x14 – 3x9 + 1 | = –2 doesn't have any solution. Fact I. |x*y| = |x|*|y|.

0 5-5

Hence | 5 | = 5 = | -5 | = distance from 5 or –5 to 0.

| -5 | = 5 | 5 | = 5

Because it is a distance measurement, |x| is nonnegative. Algebraic definition of absolute value

The geometric meaning of the absolute value of x, denoted as |x|, is the distance measured from x to 0 on the real line.

Absolute Value Equations

Because –5 is negative, use the 2nd part of the rule and take its opposite to obtain its abs. value.

|x|= x if x is positive or 0. –x (opposite of x) if x is negative.{

Hence | –5 | = –(–5) = 5. Since the absolute value is never negative, an equation such as |x14 – 3x9 + 1 | = –2 doesn't have any solution. Fact I. |x*y| = |x|*|y|. For example, |–2*3 | = |–2|*|3| = 6.

0 5-5

Hence | 5 | = 5 = | -5 | = distance from 5 or –5 to 0.

| -5 | = 5 | 5 | = 5

Because it is a distance measurement, |x| is nonnegative. Algebraic definition of absolute value

The geometric meaning of the absolute value of x, denoted as |x|, is the distance measured from x to 0 on the real line.

Because –5 is negative, use the 2nd part of the rule and take its opposite to obtain its abs. value.

Absolute Value Equations

|x|= x if x is positive or 0. –x (opposite of x) if x is negative.{

Hence | –5 | = –(–5) = 5. Since the absolute value is never negative, an equation such as |x14 – 3x9 + 1 | = –2 doesn't have any solution. Fact I. |x*y| = |x|*|y|. For example, |–2*3 | = |–2|*|3| = 6.Warning: In general |x ± y| |x| ± |y|.

0 5-5

Hence | 5 | = 5 = | -5 | = distance from 5 or –5 to 0.

| -5 | = 5 | 5 | = 5

Because it is a distance measurement, |x| is nonnegative. Algebraic definition of absolute value

The geometric meaning of the absolute value of x, denoted as |x|, is the distance measured from x to 0 on the real line.

Absolute Value Equations

Because –5 is negative, use the 2nd part of the rule and take its opposite to obtain its abs. value.

|x|= x if x is positive or 0. –x (opposite of x) if x is negative.{

Hence | –5 | = –(–5) = 5. Since the absolute value is never negative, an equation such as |x14 – 3x9 + 1 | = –2 doesn't have any solution. Fact I. |x*y| = |x|*|y|. For example, |–2*3 | = |–2|*|3| = 6.Warning: In general |x ± y| |x| ± |y|.For instance, |2 – 3 | |2| – |3| |2| + |3|.

0 5-5

Hence | 5 | = 5 = | -5 | = distance from 5 or –5 to 0.

| -5 | = 5 | 5 | = 5

Because it is a distance measurement, |x| is nonnegative. Algebraic definition of absolute value

The geometric meaning of the absolute value of x, denoted as |x|, is the distance measured from x to 0 on the real line.

Absolute Value Equations

Because –5 is negative, use the 2nd part of the rule and take its opposite to obtain its abs. value.

|x|= x if x is positive or 0. –x (opposite of x) if x is negative.{

Hence | –5 | = –(–5) = 5. Since the absolute value is never negative, an equation such as |x14 – 3x9 + 1 | = –2 doesn't have any solution. Fact I. |x*y| = |x|*|y|. For example, |–2*3 | = |–2|*|3| = 6.Warning: In general |x ± y| |x| ± |y|.For instance, |2 – 3 | |2| – |3| |2| + |3|.

0 5-5

Hence | 5 | = 5 = | -5 | = distance from 5 or –5 to 0.

| -5 | = 5 | 5 | = 5

Because it is a distance measurement, |x| is nonnegative. Algebraic definition of absolute value

The geometric meaning of the absolute value of x, denoted as |x|, is the distance measured from x to 0 on the real line.

A “| |” can not be split into two | |’s when adding or subtracting.

Absolute Value Equations

Because –5 is negative, use the 2nd part of the rule and take its opposite to obtain its abs. value.

Because |x±y| |x|±|y|, many steps for solving regular equations are not valid for | |-equations.

Absolute Value Equations

Because |x±y| |x|±|y|, many steps for solving regular equations are not valid for | |-equations. We have to solve an | |-equation by rephrasing it into two equations without the | |.

Absolute Value Equations

Fact II: If |#| = a, a >0, (where # is any expression)

Because |x±y| |x|±|y|, many steps for solving regular equations are not valid for | |-equations. We have to solve an | |-equation by rephrasing it into two equations without the | |.

Absolute Value Equations

Fact II: If |#| = a, a >0, (where # is any expression) then # = –a or # = a.

Because |x±y| |x|±|y|, many steps for solving regular equations are not valid for | |-equations. We have to solve an | |-equation by rephrasing it into two equations without the | |.

0 a–a

Absolute Value Equations

Fact II: If |#| = a, a >0, (where # is any expression) then # = –a or # = a. Example A. Solve for x.a. | x | = 3

Because |x±y| |x|±|y|, many steps for solving regular equations are not valid for | |-equations. We have to solve an | |-equation by rephrasing it into two equations without the | |.

0 a–a

Absolute Value Equations

Fact II: If |#| = a, a >0, (where # is any expression) then # = –a or # = a. Example A. Solve for x.a. | x | = 3 then x = –3 or x = 3

Because |x±y| |x|±|y|, many steps for solving regular equations are not valid for | |-equations. We have to solve an | |-equation by rephrasing it into two equations without the | |.

0 a–a

Absolute Value Equations

Fact II: If |#| = a, a >0, (where # is any expression) then # = –a or # = a. Example A. Solve for x.a. | x | = 3 then x = –3 or x = 3

b. | –2x | = 5

–2x = –5 or –2x = 5 x = 5/2

In picture, if | x | = 3 then

0 x = 3 x = –3

Because |x±y| |x|±|y|, many steps for solving regular equations are not valid for | |-equations. We have to solve an | |-equation by rephrasing it into two equations without the | |.

0 a–a

Dropping the “| |” and set the formula to 5 and –5.

Absolute Value Equations

Fact II: If |#| = a, a >0, (where # is any expression) then # = –a or # = a. Example A. Solve for x.a. | x | = 3 then x = –3 or x = 3

Because |x±y| |x|±|y|, many steps for solving regular equations are not valid for | |-equations. We have to solve an | |-equation by rephrasing it into two equations without the | |.

0 a–a

Absolute Value Equations

Fact II: If |#| = a, a >0, (where # is any expression) then # = –a or # = a. Example A. Solve for x.a. | x | = 3 then x = –3 or x = 3

In picture, if | x | = 3 then

0 x = 3 x = –3

Because |x±y| |x|±|y|, many steps for solving regular equations are not valid for | |-equations. We have to solve an | |-equation by rephrasing it into two equations without the | |.

0 a–a

Absolute Value Equations

Fact II: If |#| = a, a >0, (where # is any expression) then # = –a or # = a. Example A. Solve for x.a. | x | = 3 then x = –3 or x = 3

b. | –2x | = 5

In picture, if | x | = 3 then

0 x = 3 x = –3

Because |x±y| |x|±|y|, many steps for solving regular equations are not valid for | |-equations. We have to solve an | |-equation by rephrasing it into two equations without the | |.

0 a–a

Absolute Value Equations

Fact II: If |#| = a, a >0, (where # is any expression) then # = –a or # = a. Example A. Solve for x.a. | x | = 3 then x = –3 or x = 3

b. | –2x | = 5

In picture, if | x | = 3 then

0 x = 3 x = –3

Because |x±y| |x|±|y|, many steps for solving regular equations are not valid for | |-equations. We have to solve an | |-equation by rephrasing it into two equations without the | |.

0 a–a

Dropping the “| |” and set the formula to 5 and –5.

Absolute Value Equations

Fact II: If |#| = a, a >0, (where # is any expression) then # = –a or # = a. Example A. Solve for x.a. | x | = 3 then x = –3 or x = 3

b. | –2x | = 5

–2x = –5 or –2x = 5

In picture, if | x | = 3 then

0 x = 3 x = –3

Dropping the “| |” and set the formula to 5 and –5.

Because |x±y| |x|±|y|, many steps for solving regular equations are not valid for | |-equations. We have to solve an | |-equation by rephrasing it into two equations without the | |.

0 a–a

Absolute Value Equations

Fact II: If |#| = a, a >0, (where # is any expression) then # = –a or # = a. Example A. Solve for x.a. | x | = 3 then x = –3 or x = 3

b. | –2x | = 5

–2x = –5 or –2x = 5 x = –5/2So x = 5/2

In picture, if | x | = 3 then

0 x = 3 x = –3

Because |x±y| |x|±|y|, many steps for solving regular equations are not valid for | |-equations. We have to solve an | |-equation by rephrasing it into two equations without the | |.

0 a–a

Dropping the “| |” and set the formula to 5 and –5.

Absolute Value Equations

c. | 2x – 3 | = 5Absolute Value Equations

c. | 2x – 3 | = 5Absolute Value Equations

Remember that |2x– 3 | |2x| – |3|

c. | 2x – 3 | = 5Drop the “| |”.

Absolute Value EquationsRemember that |2x– 3 | |2x| – |3|

c. | 2x – 3 | = 5

2x – 3 = –5 or 2x – 3 = 5 Drop the “| |”. Settings

Absolute Value EquationsRemember that |2x– 3 | |2x| – |3|

c. | 2x – 3 | = 5

2x – 3 = –5 or 2x – 3 = 5 2x = –2

Drop the “| |”. Settings

Absolute Value EquationsRemember that |2x– 3 | |2x| – |3|

c. | 2x – 3 | = 5

2x – 3 = –5 or 2x – 3 = 5 2x = –2 x = –1

Drop the “| |”. Settings

Absolute Value EquationsRemember that |2x– 3 | |2x| – |3|

So

c. | 2x – 3 | = 5

2x – 3 = –5 or 2x – 3 = 5 2x = 82x = –2

x = –1

Drop the “| |”. Settings

Absolute Value EquationsRemember that |2x– 3 | |2x| – |3|

So

c. | 2x – 3 | = 5

2x – 3 = –5 or 2x – 3 = 5 2x = 8

x = 42x = –2 x = –1

Drop the “| |”. Settings

Absolute Value EquationsRemember that |2x– 3 | |2x| – |3|

So

d. | 2 – 3x | + 2 = 4

c. | 2x – 3 | = 5

2x – 3 = –5 or 2x – 3 = 5 2x = 8

x = 42x = –2 x = –1

Drop the “| |”. Settings

Absolute Value EquationsRemember that |2x– 3 | |2x| – |3|

So

d. | 2 – 3x | + 2 = 4We have to isolate the | |-term before the dropping the “| |”.

c. | 2x – 3 | = 5

2x – 3 = –5 or 2x – 3 = 5 2x = 8

x = 42x = –2 x = –1

Drop the “| |”. Settings

Absolute Value EquationsRemember that |2x– 3 | |2x| – |3|

So

d. | 2 – 3x | + 2 = 4We have to isolate the | |-term before the dropping the “| |”.

c. | 2x – 3 | = 5

2x – 3 = –5 or 2x – 3 = 5 2x = 8

x = 42x = –2 x = –1

Drop the “| |”. Settings

Incorrect versions:2–3x+2=–4 or 2–3x+2=4

Absolute Value EquationsRemember that |2x– 3 | |2x| – |3|

So

d. | 2 – 3x | + 2 = 4We have to isolate the | |-term before the dropping the “| |”.| 2 – 3x | + 2 = 4

c. | 2x – 3 | = 5

2x – 3 = –5 or 2x – 3 = 5 2x = 8

x = 42x = –2 x = –1

Drop the “| |”. Settings

Incorrect versions:2–3x+2=–4 or 2–3x+2=4

Absolute Value EquationsRemember that |2x– 3 | |2x| – |3|

So

d. | 2 – 3x | + 2 = 4We have to isolate the | |-term before the dropping the “| |”.| 2 – 3x | + 2 = 4| 2 – 3x | = 2

c. | 2x – 3 | = 5

2x – 3 = –5 or 2x – 3 = 5 2x = 8

x = 42x = –2 x = –1

Drop the “| |”. Settings

Incorrect versions:2–3x+2=–4 or 2–3x+2=4

Absolute Value EquationsRemember that |2x– 3 | |2x| – |3|

So

d. | 2 – 3x | + 2 = 4We have to isolate the | |-term before the dropping the “| |”.| 2 – 3x | + 2 = 4| 2 – 3x | = 2Drop the | |.

c. | 2x – 3 | = 5

2x – 3 = –5 or 2x – 3 = 5 2x = 8

x = 42x = –2 x = –1

Drop the “| |”. Settings

Incorrect versions:2–3x+2=–4 or 2–3x+2=4

Absolute Value EquationsRemember that |2x– 3 | |2x| – |3|

So

d. | 2 – 3x | + 2 = 4We have to isolate the | |-term before the dropping the “| |”.| 2 – 3x | + 2 = 4| 2 – 3x | = 2Drop the | |. Settings

2 – 3x = 22 – 3x = –2 or

c. | 2x – 3 | = 5

2x – 3 = –5 or 2x – 3 = 5 2x = 8

x = 42x = –2 x = –1

Drop the “| |”. Settings

Incorrect versions:2–3x+2=–4 or 2–3x+2=4

Absolute Value EquationsRemember that |2x– 3 | |2x| – |3|

So

d. | 2 – 3x | + 2 = 4We have to isolate the | |-term before the dropping the “| |”.| 2 – 3x | + 2 = 4| 2 – 3x | = 2Drop the | |. Settings

2 – 3x = 22 – 3x = –2–3x = –4

or

c. | 2x – 3 | = 5

2x – 3 = –5 or 2x – 3 = 5 2x = 8

x = 42x = –2 x = –1

Drop the “| |”. Settings

Incorrect versions:2–3x+2=–4 or 2–3x+2=4

Absolute Value EquationsRemember that |2x– 3 | |2x| – |3|

So

d. | 2 – 3x | + 2 = 4We have to isolate the | |-term before the dropping the “| |”.| 2 – 3x | + 2 = 4| 2 – 3x | = 2Drop the | |. Settings

2 – 3x = 22 – 3x = –2–3x = –4

or

x = 4/3

c. | 2x – 3 | = 5

2x – 3 = –5 or 2x – 3 = 5 2x = 8

x = 42x = –2 x = –1

Drop the “| |”. Settings

Incorrect versions:2–3x+2=–4 or 2–3x+2=4

Absolute Value EquationsRemember that |2x– 3 | |2x| – |3|

So

So

d. | 2 – 3x | + 2 = 4We have to isolate the | |-term before the dropping the “| |”.| 2 – 3x | + 2 = 4| 2 – 3x | = 2Drop the | |. Settings

2 – 3x = 22 – 3x = –2–3x = 0

–3x = –4or

x = 4/3

c. | 2x – 3 | = 5

2x – 3 = –5 or 2x – 3 = 5 2x = 8

x = 42x = –2 x = –1

Drop the “| |”. Settings

Incorrect versions:2–3x+2=–4 or 2–3x+2=4

Absolute Value EquationsRemember that |2x– 3 | |2x| – |3|

So

So

d. | 2 – 3x | + 2 = 4We have to isolate the | |-term before the dropping the “| |”.| 2 – 3x | + 2 = 4| 2 – 3x | = 2Drop the | |. Settings

2 – 3x = 22 – 3x = –2–3x = 0

–3x = –4or

x = 4/3

c. | 2x – 3 | = 5

2x – 3 = –5 or 2x – 3 = 5 2x = 8

x = 42x = –2 x = –1

Drop the “| |”. Settings

Incorrect versions:2–3x+2=–4 or 2–3x+2=4

Absolute Value EquationsRemember that |2x– 3 | |2x| – |3|

x = 0 So

So

The geometric meaning of | x | as “distance from 0 and x” may be extended to the meaning for |x – y|.

Absolute Value Equations

The geometric meaning of | x | as “distance from 0 and x” may be extended to the meaning for |x – y|. The geometric meaning of |x – y| = |y – x| = “distance between x and y”.

Absolute Value Equations

yx |x – y| = |y – x|

The geometric meaning of | x | as “distance from 0 and x” may be extended to the meaning for |x – y|. The geometric meaning of |x – y| = |y – x| = “distance between x and y”. Note the two versions reflect that “distance” is always mutual in the sense that “going from x to y” is as much as “going from y to x”.

Absolute Value Equations

yx

same distance

|x – y| = |y – x|

The geometric meaning of | x | as “distance from 0 and x” may be extended to the meaning for |x – y|. The geometric meaning of |x – y| = |y – x| = “distance between x and y”. Note the two versions reflect that “distance” is always mutual in the sense that “going from x to y” is as much as “going from y to x”.

Absolute Value Equations

|9 – 3| = 6 = |3 – 9|

yx

same distance

|x – y| = |y – x|

The geometric meaning of | x | as “distance from 0 and x” may be extended to the meaning for |x – y|. The geometric meaning of |x – y| = |y – x| = “distance between x and y”. Note the two versions reflect that “distance” is always mutual in the sense that “going from x to y” is as much as “going from y to x”.

Absolute Value Equations

|9 – 3| = 6 = |3 – 9| and |(–8) – (–1)| = 7.

yx

same distance

|x – y| = |y – x|

The geometric meaning of | x | as “distance from 0 and x” may be extended to the meaning for |x – y|. The geometric meaning of |x – y| = |y – x| = “distance between x and y”. Note the two versions reflect that “distance” is always mutual in the sense that “going from x to y” is as much as “going from y to x”.

Absolute Value Equations

|9 – 3| = 6 = |3 – 9| and |(–8) – (–1)| = 7. We can solve for x using this method.

yx

same distance

|x – y| = |y – x|

The geometric meaning of | x | as “distance from 0 and x” may be extended to the meaning for |x – y|. The geometric meaning of |x – y| = |y – x| = “distance between x and y”. Note the two versions reflect that “distance” is always mutual in the sense that “going from x to y” is as much as “going from y to x”.

Absolute Value Equations

|9 – 3| = 6 = |3 – 9| and |(–8) – (–1)| = 7. We can solve for x using this method. Example B. Solve for x geometrically if |x – 7| = 12.

yx

same distance

|x – y| = |y – x|

The geometric meaning of | x | as “distance from 0 and x” may be extended to the meaning for |x – y|. The geometric meaning of |x – y| = |y – x| = “distance between x and y”. Note the two versions reflect that “distance” is always mutual in the sense that “going from x to y” is as much as “going from y to x”.

Absolute Value Equations

|9 – 3| = 6 = |3 – 9| and |(–8) – (–1)| = 7. We can solve for x using this method. Example B. Solve for x geometrically if |x – 7| = 12.We want the locations of x's that are 12 units away from 7.

yx

same distance

|x – y| = |y – x|

The geometric meaning of | x | as “distance from 0 and x” may be extended to the meaning for |x – y|. The geometric meaning of |x – y| = |y – x| = “distance between x and y”. Note the two versions reflect that “distance” is always mutual in the sense that “going from x to y” is as much as “going from y to x”.

Absolute Value Equations

|9 – 3| = 6 = |3 – 9| and |(–8) – (–1)| = 7. We can solve for x using this method. Example B. Solve for x geometrically if |x – 7| = 12.We want the locations of x's that are 12 units away from 7.

7

1212

x x

yx

same distance

|x – y| = |y – x|

The geometric meaning of | x | as “distance from 0 and x” may be extended to the meaning for |x – y|. The geometric meaning of |x – y| = |y – x| = “distance between x and y”. Note the two versions reflect that “distance” is always mutual in the sense that “going from x to y” is as much as “going from y to x”.

Absolute Value Equations

|9 – 3| = 6 = |3 – 9| and |(–8) – (–1)| = 7. We can solve for x using this method.

yx

same distance

Example B. Solve for x geometrically if |x – 7| = 12.We want the locations of x's that are 12 units away from 7.

7

1212

x = – 5 xSo to the left x = 7 – 12 = – 5,

|x – y| = |y – x|

The geometric meaning of | x | as “distance from 0 and x” may be extended to the meaning for |x – y|. The geometric meaning of |x – y| = |y – x| = “distance between x and y”. Note the two versions reflect that “distance” is always mutual in the sense that “going from x to y” is as much as “going from y to x”.

Absolute Value Equations

|9 – 3| = 6 = |3 – 9| and |(–8) – (–1)| = 7. We can solve for x using this method.

yx

same distance

Example B. Solve for x geometrically if |x – 7| = 12.We want the locations of x's that are 12 units away from 7.

7

1212

x = 19

So to the left x = 7 – 12 = – 5, and to the right x = 7 + 12 = 19.

|x – y| = |y – x|

x = – 5

The rule for dropping the | | extends to the following setups. Absolute Value Equations

Fact III: If |E1| = |E2| where E1, E2 are two expressions, The rule for dropping the | | extends to the following setups.

Absolute Value Equations

Fact III: If |E1| = |E2| where E1, E2 are two expressions, then E1 = E2 or E1 = –(E2).

Absolute Value EquationsThe rule for dropping the | | extends to the following setups.

Fact III: If |E1| = |E2| where E1, E2 are two expressions, then E1 = E2 or E1 = –(E2).Example C. Solve for x.a. |2x – 3| = |3x + 1|

Absolute Value EquationsThe rule for dropping the | | extends to the following setups.

Fact III: If |E1| = |E2| where E1, E2 are two expressions, then E1 = E2 or E1 = –(E2).Example C. Solve for x.

Dropping the “| |” and seta. |2x – 3| = |3x + 1|

Absolute Value EquationsThe rule for dropping the | | extends to the following setups.

Fact III: If |E1| = |E2| where E1, E2 are two expressions, then E1 = E2 or E1 = –(E2).Example C. Solve for x.

Dropping the “| |” and seta. |2x – 3| = |3x + 1|

2x – 3 = 3x + 1 or 2x – 3 = –(3x + 1)

Absolute Value EquationsThe rule for dropping the | | extends to the following setups.

Fact III: If |E1| = |E2| where E1, E2 are two expressions, then E1 = E2 or E1 = –(E2).Example C. Solve for x.

Dropping the “| |” and seta. |2x – 3| = |3x + 1|

2x – 3 = 3x + 1 or 2x – 3 = –(3x + 1)–1 – 3 = 3x – 2x

Absolute Value EquationsThe rule for dropping the | | extends to the following setups.

Fact III: If |E1| = |E2| where E1, E2 are two expressions, then E1 = E2 or E1 = –(E2).Example C. Solve for x.

Dropping the “| |” and set

–4 = x

a. |2x – 3| = |3x + 1|

2x – 3 = 3x + 1 or 2x – 3 = –(3x + 1)–1 – 3 = 3x – 2x

Absolute Value EquationsThe rule for dropping the | | extends to the following setups.

So

Fact III: If |E1| = |E2| where E1, E2 are two expressions, then E1 = E2 or E1 = –(E2).Example C. Solve for x.

Dropping the “| |” and set

–4 = x

a. |2x – 3| = |3x + 1|

2x – 3 = 3x + 1 or 2x – 3 = –(3x + 1)2x – 3 = –3x – 1 –1 – 3 = 3x – 2x

Absolute Value EquationsThe rule for dropping the | | extends to the following setups.

So

Fact III: If |E1| = |E2| where E1, E2 are two expressions, then E1 = E2 or E1 = –(E2).Example C. Solve for x.

Dropping the “| |” and set

–4 = x 5x = 2

a. |2x – 3| = |3x + 1|

2x – 3 = 3x + 1 or 2x – 3 = –(3x + 1)2x – 3 = –3x – 1 –1 – 3 = 3x – 2x

Absolute Value EquationsThe rule for dropping the | | extends to the following setups.

So

Fact III: If |E1| = |E2| where E1, E2 are two expressions, then E1 = E2 or E1 = –(E2).Example C. Solve for x.

Dropping the “| |” and set

–4 = x 5x = 2

a. |2x – 3| = |3x + 1|

2x – 3 = 3x + 1 or 2x – 3 = –(3x + 1)2x – 3 = –3x – 1

x = 2/5

–1 – 3 = 3x – 2x

Absolute Value EquationsThe rule for dropping the | | extends to the following setups.

So

Fact III: If |E1| = |E2| where E1, E2 are two expressions, then E1 = E2 or E1 = –(E2).Example C. Solve for x.

Dropping the “| |” and set

–4 = x 5x = 2

b. |x – 1| = |x + 1|

a. |2x – 3| = |3x + 1|

2x – 3 = 3x + 1 or 2x – 3 = –(3x + 1)2x – 3 = –3x – 1

x = 2/5

–1 – 3 = 3x – 2x

Absolute Value EquationsThe rule for dropping the | | extends to the following setups.

So

Fact III: If |E1| = |E2| where E1, E2 are two expressions, then E1 = E2 or E1 = –(E2).Example C. Solve for x.

Dropping the “| |” and set

–4 = x 5x = 2

b. |x – 1| = |x + 1|

a. |2x – 3| = |3x + 1|

2x – 3 = 3x + 1 or 2x – 3 = –(3x + 1)2x – 3 = –3x – 1

x = 2/5

Dropping the “| |” and set

–1 – 3 = 3x – 2x

Absolute Value EquationsThe rule for dropping the | | extends to the following setups.

So

Fact III: If |E1| = |E2| where E1, E2 are two expressions, then E1 = E2 or E1 = –(E2).Example C. Solve for x.

Dropping the “| |” and set

–4 = x 5x = 2

b. |x – 1| = |x + 1|

a. |2x – 3| = |3x + 1|

2x – 3 = 3x + 1 or 2x – 3 = –(3x + 1)2x – 3 = –3x – 1

x = 2/5

Dropping the “| |” and setx – 1 = x + 1

–1 – 3 = 3x – 2x

x – 1 = –(x + 1) or

Absolute Value EquationsThe rule for dropping the | | extends to the following setups.

So

Fact III: If |E1| = |E2| where E1, E2 are two expressions, then E1 = E2 or E1 = –(E2).Example C. Solve for x.

Dropping the “| |” and set

–4 = x 5x = 2

b. |x – 1| = |x + 1|

a. |2x – 3| = |3x + 1|

2x – 3 = 3x + 1 or 2x – 3 = –(3x + 1)2x – 3 = –3x – 1

x = 2/5

Dropping the “| |” and setx – 1 = x + 1

0 = 2

–1 – 3 = 3x – 2x

x – 1 = –(x + 1) or

Absolute Value EquationsThe rule for dropping the | | extends to the following setups.

So

Fact III: If |E1| = |E2| where E1, E2 are two expressions, then E1 = E2 or E1 = –(E2).Example C. Solve for x.

Dropping the “| |” and set

–4 = x 5x = 2

b. |x – 1| = |x + 1|

a. |2x – 3| = |3x + 1|

2x – 3 = 3x + 1 or 2x – 3 = –(3x + 1)

x – 1 = –(x + 1)

2x – 3 = –3x – 1

x = 2/5

Dropping the “| |” and setor x – 1 = x + 1

0 = 2

–1 – 3 = 3x – 2x

Impossible!

Absolute Value EquationsThe rule for dropping the | | extends to the following setups.

So

Fact III: If |E1| = |E2| where E1, E2 are two expressions, then E1 = E2 or E1 = –(E2).Example C. Solve for x.

Dropping the “| |” and set

–4 = x 5x = 2

b. |x – 1| = |x + 1|

a. |2x – 3| = |3x + 1|

2x – 3 = 3x + 1 or 2x – 3 = –(3x + 1)

x – 1 = –(x + 1)

2x – 3 = –3x – 1

x = 2/5

Dropping the “| |” and setor x – 1 = x + 1

x – 1 = –x – 1 0 = 2

–1 – 3 = 3x – 2x

Impossible!

Absolute Value EquationsThe rule for dropping the | | extends to the following setups.

So

Fact III: If |E1| = |E2| where E1, E2 are two expressions, then E1 = E2 or E1 = –(E2).Example C. Solve for x.

Dropping the “| |” and set

–4 = x 5x = 2

b. |x – 1| = |x + 1|

a. |2x – 3| = |3x + 1|

2x – 3 = 3x + 1 or 2x – 3 = –(3x + 1)

x – 1 = –(x + 1)

2x – 3 = –3x – 1

x = 2/5

Dropping the “| |” and setor x – 1 = x + 1

x – 1 = –x – 1 2x = 0 x = 0

0 = 2

–1 – 3 = 3x – 2x

Impossible!

Absolute Value EquationsThe rule for dropping the | | extends to the following setups.

So

Ex. A. 1. Is it always true that I+x| = x? Give reason for your answer. 2. Is it always true that |–x| = x? Give reason for your answer.

Absolute Value Equations

Ex. B. Drop the | | and write the problem into two equations then solve for x (if any) and label the answer(s) on the real line.3. |x| = 2 4. |x| = 5 5. |–x| = 2 6. |–x| = 5

7. |x| = –2 8. |–2x| = 6 9. |–3x| = 6 10. |–x| = –5

11. |3 – x| = –5 12. |3 + x| = 7 13. |x – 9| = 5

14. |5 – x| = 5 15. |4 + x| = 9 16. |2x + 1| = 3

17. |4 – 5x| = 3 18. |3 + 2x| = 7 19. |–2x + 3| = 5

20. |4 – 5x| = –3 21. |2x + 1| – 1= 5 21. 3|2x + 1| – 1= 5

Absolute Value EquationsEx. C. Solve for x by using the geometric method.

28. |4 – 5x| = |3 + 2x|

30. |4 – 5x| = |2x + 1| 31. |3x + 1| = |5 – x|

22. |x – 2| = 1 23. |3 – x| = 5 24. |x – 5| = 5

25. |7 – x| = 3 26. |8 + x| = 9 27. |x + 1| = 3

Ex. D. Drop the | | then solve for x.

29. |–2x + 3|= |3 – 2x|

32. |3 – 2x| = |2x + 1| 33. |3x + 1| = |–3x – 1|