Lesson Solving numerical and linear absolute value equations

Source code of 'Solving numerical and linear absolute value equations'

This Lesson (Solving numerical and linear absolute value equations) was created by by lyra(94)

: View Source, Show
About lyra: I am able to help with Algebra I and Algebra II problems. I Specialize in equations, counting and probability, inequalities and systems of equations. I also have first hand experience with the JHU Honors Algebra II course and curriculum.

&#56256;&#56490;The absolute value sign is: {{{abs(x)}}}
&#56256;&#56490; The absolute value of any number represents its positive distance from 0 on a number line.  Remember, there can never be a negative distance between objects, or numbers!  {{{abs(x)=-y}}} is not possible!!!
&#56256;&#56490; If the number inside the absolute value  is positive or equal to 0, its absolute
value is equal to that number.  For example {{{abs(4)=4}}}
&#56256;&#56490; If the number inside the absolute value is negative, then its absolute value is the negative of itself.  For example {{{abs(-7)=-1(-7)=7}}}
&#56256;&#56490;Absolute Value is just a measure of distance
&#56256;&#56490;Absolute value can be used with equations containing variables.  Example: {{{abs(2x+5)=2}}}
&#56256;&#56490;If an absolute value equation is equal to 0, you can remove the absolute value and work with the equation alone.  For example: {{{abs(5x+3)=0}}} is the same as {{{5x+3=0}}} since {{{+-0}}} is just 0.

Now for some examples of how to solve absolute value expressions:

1. {{{abs(-1/3)=-1(-1/3)=1/3}}}

2. {{{abs(5-3x)=7}}}
First equation: 
{{{5-3x=7}}}  Setting inside linear expression = to 7 
{{{5-5-3x=7-5}}}  Get 3x alone on LH side by eliminating constant on LH side
{{{-3x=2}}} Condensing above equation
{{{-3x/3=2/-3}}}  Divide by -3 to get x alone on LH side
{{{x=-2/3}}} Answer to first equation 
Second equation: 
{{{5-3x=-7}}}  Setting inside linear expression = to -7 
{{{5-5-3x=-7-5}}}  Get 3x alone on LH side by eliminating constant on LH side
{{{-3x=-12}}} Condensing above equation
{{{-3x/-3=-12/-3}}}  Divide by -3 to get x alone on LH side
{{{x=4}}} Answer to second equation

Thus, the answer to example 2 is x={{{-2/3}}} or x={{{4}}}

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Now try some on your own:

1. What is the {{{abs(-1)}}}?

2. What is the {{{abs(-73/154)}}}?

3. What is the {{{abs(pi)}}}?

4. What is the {{{abs(infinity)}}}?

5. What is the {{{abs(sqrt(2))}}}?

6. What is the {{{abs(2x-1)=5}}}?

7. What is the {{{abs(10x+5)=-5}}}?

8. What is the {{{abs(2x-4)=0}}}?

9. What is the {{{abs(7x+2)=10}}}?

10.What is the {{{abs(3x-5)=-1}}}?

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Answers to problems

1. What is the {{{abs(-1)=-1*-1=1}}}?

2. What is the {{{abs(-73/154)=-1*-73/154=73/154}}}?

3. What is the {{{abs(pi)= pi}}}?

4. What is the {{{abs(infinity)= infinity}}}?

5. What is the {{{abs(sqrt(2))= sqrt(2))}}}?

6. What is the {{{abs(2x-1)=5}}}?
Either {{{2x-1=5}}} or {{{2x-1=-5}}}
If {{{2x-1=5}}}
{{{2x-1+1=5+1}}}  Add 1 to both sides to eliminate constant on LH side and get 2x alone on LH side
{{{2x=6}}}  Simplify expression above
{{{2x/2=6/2}}}  Divide by 2 to get x alone on LH side
{{{x=3}}}  Answer to equation 1

If {{{2x-1=-5}}}
{{{2x-1+1=-5+1}}}  Add 1 to both sides to eliminate constant on LH side and get 2x alone on LH side
{{{2x=-4}}}  Simplify expression above
{{{2x/2=-4/2}}}  Divide by 2 to get x alone on LH side
{{{x=-2}}}  Answer to equation 2

Thus, the answer is x={{{3}}} or x={{{-2}}}

7. What is the {{{abs(10x+5)=-5}}}?

This is a trick!  No absolute value can be negative, it is not possible for there to be a negitive distance between something!

8. What is the {{{abs(2x-4)=0}}}?

{{{-0}}} is just 0, so we just solve with the equation 2x-4=0

{{{2x-4+4=0+4}}}  Add 4 to both sides to eliminate constant on LH side and get 2x alone on LH side
{{{2x=4}}}  Simplify expression above
{{{2x/2=4/2}}}  Divide by 2 to get x alone on LH side
{{{x=2}}}  Answer

Thus, the answer is x={{{2}}}
 
9. What is the {{{abs(7x+2)=10}}}?
Either {{{7x+2=10}}} or {{{7x+2=-10}}}
If {{{7x+2=10}}}
{{{7x+2-2=10-2}}}  Subtract 2 from both sides to eliminate constant on LH side and get 7x alone on LH side
{{{7x=8}}}  Simplify expression above
{{{7x/7=8/7}}}  Divide by 7 to get x alone on LH side
{{{x=8/7}}}  Answer to equation 1

If {{{7x+2=-10}}}
{{{7x+2-2=-10-2}}}  Subtract 2 from both sides to eliminate constant on LH side and get 2x alone on LH side
{{{7x=-12}}}  Simplify expression above
{{{7x/7=-12/7}}}  Divide by 7 to get x alone on LH side
{{{x=-12/7}}}  Answer to equation 2

Thus, the answer is x={{{8/7}}} or x={{{-12/7}}}

10.What is the {{{abs(3x-5)=-1}}}?
This is a trick!  No absolute value can be negative, it is not possible for there to be a negitive distance between something!

If you would like to learn about how to solve quadratic absolute value equations, please see my lesson "Solving quadratic absolute value equations" which will be avaliable soon.

Good luck!