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Solution - Absolute value equations

Exact form:

t = - \frac{1}{2}, \frac{1}{4}

t=-\frac{1}{2} , \frac{1}{4}

Decimal form:

t = - 0.5, 0.25

t=-0.5 , 0.25

See steps

Step-by-step explanation

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

$| t - 1 | - 3 | t | = 0$

Add $3 | t |$ to both sides of the equation:

$| t - 1 | - 3 | t | + 3 | t | = 3 | t |$

Simplify the arithmetic

$| t - 1 | = 3 | t |$

2. Rewrite the equation without absolute value bars

Use the rules:
$| x | = | y |$ → $x = \pm y$ and $| x | = | y |$ → $\pm x = y$
to write all four options of the equation
$| t - 1 | = 3 | t |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| t - 1 \| = 3 \| t \|$
$x = + y$	$(t - 1) = 3 (t)$
$x = - y$	$(t - 1) = 3 (- (t))$
$+ x = y$	$(t - 1) = 3 (t)$
$- x = y$	$- (t - 1) = 3 (t)$

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:

$\| x \| = \| y \|$	$\| t - 1 \| = 3 \| t \|$
$x = + y$ , $+ x = y$	$(t - 1) = 3 (t)$
$x = - y$ , $- x = y$	$(t - 1) = 3 (- (t))$

3. Solve the two equations for t

10 additional steps

$(t - 1) = 3 t$

Subtract from both sides:

$(t - 1) - 3 t = (3 t) - 3 t$

Group like terms:

$(t - 3 t) - 1 = (3 t) - 3 t$

Simplify the arithmetic:

$- 2 t - 1 = (3 t) - 3 t$

Simplify the arithmetic:

$- 2 t - 1 = 0$

Add to both sides:

$(- 2 t - 1) + 1 = 0 + 1$

Simplify the arithmetic:

$- 2 t = 0 + 1$

Simplify the arithmetic:

$- 2 t = 1$

Divide both sides by :

$\frac{(- 2 t)}{- 2} = \frac{1}{- 2}$

Cancel out the negatives:

$\frac{2 t}{2} = \frac{1}{- 2}$

Simplify the fraction:

$t = \frac{1}{- 2}$

Move the negative sign from the denominator to the numerator:

$t = \frac{- 1}{2}$

10 additional steps

$(t - 1) = 3 \cdot - t$

Group like terms:

$(t - 1) = (3 \cdot - 1) t$

Multiply the coefficients:

$(t - 1) = - 3 t$

Add to both sides:

$(t - 1) + 3 t = (- 3 t) + 3 t$

Group like terms:

$(t + 3 t) - 1 = (- 3 t) + 3 t$

Simplify the arithmetic:

$4 t - 1 = (- 3 t) + 3 t$

Simplify the arithmetic:

$4 t - 1 = 0$

Add to both sides:

$(4 t - 1) + 1 = 0 + 1$

Simplify the arithmetic:

$4 t = 0 + 1$

Simplify the arithmetic:

$4 t = 1$

Divide both sides by :

$\frac{(4 t)}{4} = \frac{1}{4}$

Simplify the fraction:

$t = \frac{1}{4}$

4. List the solutions

$t = - \frac{1}{2}, \frac{1}{4}$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | t - 1 |$
$y = 3 | t |$
The equation is true where the two lines cross.

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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.

Solution - Absolute value equations

Step-by-step explanation

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

2. Rewrite the equation without absolute value bars

3. Solve the two equations for t

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

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

2. Rewrite the equation without absolute value bars

3. Solve the two equations for t

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved