SOLUTION: i) Use the definition of absolute value to solve each equation, ii) Graph the solution |2.5d + 1.8| =

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Question 1164500: i) Use the definition of absolute value to solve each equation,
ii) Graph the solution
|2.5d + 1.8| = | 7.5d + 8|
Found 2 solutions by Theo, greenestamps:
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
|x| = |y| if and only if:
x = |y| and -x = |y|
these equations are the same as:
|y| = x and |y| = -x

|y| = x if and only if:
y = x and -y = x

|y| = -x if and only if:
y = -x and -y = -x

you have 4 equations that need to be solved.
they are:
y = x and -y = x
y = -x and -y = -x

y = x and -y = -x will get you the same answer.
-y = x and y = -x will also get you the same answer.

therefore, you only have to solve for:

y = x and -y = x

how does this relate to your problem?

your problem is |2.5d + 1.8| = |7.5d + 8|
this is true if and only if:
2.5d + 1.8 = |7.5d + 8|
-(2.5d + 1.8) = |7.5d + 8|

the first of these equations is the same as |7.5d + 8| = 2.5d + 1.8
the second of these equation is the same as |7.5d + 8| = -(2.5d + 1.8)

you only need to solve the first of these equations.
you will get the same answer as you would if you solved the second of these equations.

therefore, solve for |7.5d + 8| = 2.5d + 1.8
this is true if and only if:
(7.5d + 8) = 2.5d + 1.8
-(7.5d + 8) = 2.5d + 1.8

you solve (7.5d + 8) = 2.5d + 1.8 as follows:
this becomes 7.5d + 8 = 2.5d + 1.8
subtract 2.5d from both sides of the equation and subtract 8 from both sides of the equation and simplify to get:
5d = -6.2
solve for d to get:
d = -6.2 / 5 = -1.24

you solve -(7.5d + 8) = 2.5d + 1.8 as follows:
this becomes -7.5d - 8 = 2.5d + 1.8
subtract 2.5d from both sides of this equation and add 8 to both sides of this equation and simplify to get:
-10d = 9.8
solve for d to get:
d = 9.8/-10 = -.98

your solution is that d = -1.24 or d = -.98

i used an online absolute value calculator to confirm these results.
that calculator can be found at https://www.mathpapa.com/absolute-value-equation-calculator/

feel free to use that calculator yourself to confirm the results are as i described them.

Answer by greenestamps(13209) (Show Source):
You can put this solution on YOUR website!

The equation is of the form

When an absolute value equation is of this form, you need to solve only two equations:

and

(The other two possible equations are and ; but those two equations are equivalent to the first two.)

So....

(1)

(2)

ANSWERS: d = -0.98 and d = -1.24

SOLUTION: i) Use the definition of absolute value to solve each equation, ii) Graph the solution |2.5d + 1.8| = | 7.5d + 8|