Download - 3 2 absolute value equations-x
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|