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Solution - Absolute value equations

Exact form:

t = 0, 0

t=0 , 0

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
$3 | 3 t | = | 16 t |$
without the absolute value bars:

$\| x \| = \| y \|$	$3 \| 3 t \| = \| 16 t \|$
$x = + y$	$3 (3 t) = (16 t)$
$x = - y$	$3 (3 t) = - (16 t)$
$+ x = y$	$3 (3 t) = (16 t)$
$- x = y$	$3 (- (3 t)) = (16 t)$

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:

$\| x \| = \| y \|$	$3 \| 3 t \| = \| 16 t \|$
$x = + y$ , $+ x = y$	$3 (3 t) = (16 t)$
$x = - y$ , $- x = y$	$3 (3 t) = - (16 t)$

2. Solve the two equations for t

4 additional steps

$3 \cdot 3 t = 16 t$

Multiply the coefficients:

$9 t = 16 t$

Subtract from both sides:

$(9 t) - 16 t = (16 t) - 16 t$

Simplify the arithmetic:

$- 7 t = (16 t) - 16 t$

Simplify the arithmetic:

$- 7 t = 0$

Divide both sides by the coefficient:

$t = 0$

4 additional steps

$3 \cdot 3 t = - (16 t)$

Multiply the coefficients:

$9 t = - (16 t)$

Add to both sides:

$(9 t) + 16 t = (- 16 t) + 16 t$

Simplify the arithmetic:

$25 t = (- 16 t) + 16 t$

Simplify the arithmetic:

$25 t = 0$

Divide both sides by the coefficient:

$t = 0$

3. List the solutions

$t = 0, 0$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = 3 | 3 t |$
$y = | 16 t |$
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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