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Solution - Absolute value equations

Exact form: a=14,67
a=14 , \frac{6}{7}
Decimal form: a=14,0.857
a=14 , 0.857

Other Ways to Solve

Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

Use the rules:
|x|=|y|x=±y and |x|=|y|±x=y
to write all four options of the equation
|3a+4|=|4a10|
without the absolute value bars:

|x|=|y||3a+4|=|4a10|
x=+y(3a+4)=(4a10)
x=y(3a+4)=(4a10)
+x=y(3a+4)=(4a10)
x=y(3a+4)=(4a10)

When simplified, equations x=+y and +x=y are the same and equations x=y and x=y are the same, so we end up with only 2 equations:

|x|=|y||3a+4|=|4a10|
x=+y , +x=y(3a+4)=(4a10)
x=y , x=y(3a+4)=(4a10)

2. Solve the two equations for a

10 additional steps

(3a+4)=(4a-10)

Subtract from both sides:

(3a+4)-4a=(4a-10)-4a

Group like terms:

(3a-4a)+4=(4a-10)-4a

Simplify the arithmetic:

-a+4=(4a-10)-4a

Group like terms:

-a+4=(4a-4a)-10

Simplify the arithmetic:

a+4=10

Subtract from both sides:

(-a+4)-4=-10-4

Simplify the arithmetic:

a=104

Simplify the arithmetic:

a=14

Multiply both sides by :

-a·-1=-14·-1

Remove the one(s):

a=-14·-1

Simplify the arithmetic:

a=14

10 additional steps

(3a+4)=-(4a-10)

Expand the parentheses:

(3a+4)=-4a+10

Add to both sides:

(3a+4)+4a=(-4a+10)+4a

Group like terms:

(3a+4a)+4=(-4a+10)+4a

Simplify the arithmetic:

7a+4=(-4a+10)+4a

Group like terms:

7a+4=(-4a+4a)+10

Simplify the arithmetic:

7a+4=10

Subtract from both sides:

(7a+4)-4=10-4

Simplify the arithmetic:

7a=104

Simplify the arithmetic:

7a=6

Divide both sides by :

(7a)7=67

Simplify the fraction:

a=67

3. List the solutions

a=14,67
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
y=|3a+4|
y=|4a10|
The equation is true where the two lines cross.

Why learn this

We encounter absolute values almost daily. For example: If you walk 3 miles to school, do you also walk minus 3 miles when you go back home? The answer is no because distances use absolute value. The absolute value of the distance between home and school is 3 miles, there or back.
In short, absolute values help us deal with concepts like distance, ranges of possible values, and deviation from a set value.