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Solution - Absolute value equations

Exact form:

t = - 4, \frac{4}{3}

t=-4 , \frac{4}{3}

Mixed number form:

t = - 4, 1 \frac{1}{3}

t=-4 , 1\frac{1}{3}

Decimal form:

t = - 4, 1.333

t=-4 , 1.333

See steps

Step-by-step explanation

1. 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
$| t - 4 | = | 2 t |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| t - 4 \| = \| 2 t \|$
$x = + y$	$(t - 4) = (2 t)$
$x = - y$	$(t - 4) = - (2 t)$
$+ x = y$	$(t - 4) = (2 t)$
$- x = y$	$- (t - 4) = (2 t)$

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:

$\| x \| = \| y \|$	$\| t - 4 \| = \| 2 t \|$
$x = + y$ , $+ x = y$	$(t - 4) = (2 t)$
$x = - y$ , $- x = y$	$(t - 4) = - (2 t)$

2. Solve the two equations for t

9 additional steps

$(t - 4) = 2 t$

Subtract from both sides:

$(t - 4) - 2 t = (2 t) - 2 t$

Group like terms:

$(t - 2 t) - 4 = (2 t) - 2 t$

Simplify the arithmetic:

$- t - 4 = (2 t) - 2 t$

Simplify the arithmetic:

$- t - 4 = 0$

Add to both sides:

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

Simplify the arithmetic:

$- t = 0 + 4$

Simplify the arithmetic:

$- t = 4$

Multiply both sides by :

$- t \cdot - 1 = 4 \cdot - 1$

Remove the one(s):

$t = 4 \cdot - 1$

Simplify the arithmetic:

$t = - 4$

7 additional steps

$(t - 4) = - 2 t$

Add to both sides:

$(t - 4) + 4 = (- 2 t) + 4$

Simplify the arithmetic:

$t = (- 2 t) + 4$

Add to both sides:

$t + 2 t = ((- 2 t) + 4) + 2 t$

Simplify the arithmetic:

$3 t = ((- 2 t) + 4) + 2 t$

Group like terms:

$3 t = (- 2 t + 2 t) + 4$

Simplify the arithmetic:

$3 t = 4$

Divide both sides by :

$\frac{(3 t)}{3} = \frac{4}{3}$

Simplify the fraction:

$t = \frac{4}{3}$

3. List the solutions

$t = - 4, \frac{4}{3}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | t - 4 |$
$y = | 2 t |$
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 without absolute value bars

2. Solve the two equations for t

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

2. Solve the two equations for t

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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