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Solution - Absolute value equations

Exact form: t=-52
t=-\frac{5}{2}
Mixed number form: t=-212
t=-2\frac{1}{2}
Decimal form: t=−2.5
t=-2.5
See steps

Other Ways to Solve

Absolute value equations

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

|t+6|+|t−1|=0

Add −|t−1| to both sides of the equation:

|t+6|+|t−1|−|t−1|=−|t−1|

Simplify the arithmetic

|t+6|=−|t−1|

2. 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
|t+6|=−|t−1|
without the absolute value bars:

|x|=|y||t+6|=−|t−1|
x=+y(t+6)=−(t−1)
x=−y(t+6)=−−(t−1)
+x=y(t+6)=−(t−1)
−x=y−(t+6)=−(t−1)

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||t+6|=−|t−1|
x=+y , +x=y(t+6)=−(t−1)
x=−y , −x=y(t+6)=−−(t−1)

3. Solve the two equations for t

10 additional steps

(t+6)=-(t-1)

Expand the parentheses:

(t+6)=-t+1

Add to both sides:

(t+6)+t=(-t+1)+t

Group like terms:

(t+t)+6=(-t+1)+t

Simplify the arithmetic:

2t+6=(-t+1)+t

Group like terms:

2t+6=(-t+t)+1

Simplify the arithmetic:

2t+6=1

Subtract from both sides:

(2t+6)-6=1-6

Simplify the arithmetic:

2t=1−6

Simplify the arithmetic:

2t=−5

Divide both sides by :

(2t)2=-52

Simplify the fraction:

t=-52

6 additional steps

(t+6)=-(-(t-1))

NT_MSLUS_MAINSTEP_RESOLVE_DOUBLE_MINUS:

(t+6)=t-1

Subtract from both sides:

(t+6)-t=(t-1)-t

Group like terms:

(t-t)+6=(t-1)-t

Simplify the arithmetic:

6=(t-1)-t

Group like terms:

6=(t-t)-1

Simplify the arithmetic:

6=−1

The statement is false:

6=−1

The equation is false so it has no solution.

4. List the solutions

t=-52
(1 solution(s))

5. Graph

Each line represents the function of one side of the equation:
y=|t+6|
y=−|t−1|
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.

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