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Solution - Absolute value equations

Exact form:

x = - \frac{13}{2}, - \frac{7}{10}

x=-\frac{13}{2} , -\frac{7}{10}

Mixed number form:

x = - 6 \frac{1}{2}, - \frac{7}{10}

x=-6\frac{1}{2} , -\frac{7}{10}

Decimal form:

x = - 6.5, - 0.7

x=-6.5 , -0.7

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
$| 6 x + 10 | = | 4 x - 3 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 6 x + 10 \| = \| 4 x - 3 \|$
$x = + y$	$(6 x + 10) = (4 x - 3)$
$x = - y$	$(6 x + 10) = - (4 x - 3)$
$+ x = y$	$(6 x + 10) = (4 x - 3)$
$- x = y$	$- (6 x + 10) = (4 x - 3)$

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:

$\| x \| = \| y \|$	$\| 6 x + 10 \| = \| 4 x - 3 \|$
$x = + y$ , $+ x = y$	$(6 x + 10) = (4 x - 3)$
$x = - y$ , $- x = y$	$(6 x + 10) = - (4 x - 3)$

2. Solve the two equations for x

9 additional steps

$(6 x + 10) = (4 x - 3)$

Subtract from both sides:

$(6 x + 10) - 4 x = (4 x - 3) - 4 x$

Group like terms:

$(6 x - 4 x) + 10 = (4 x - 3) - 4 x$

Simplify the arithmetic:

$2 x + 10 = (4 x - 3) - 4 x$

Group like terms:

$2 x + 10 = (4 x - 4 x) - 3$

Simplify the arithmetic:

$2 x + 10 = - 3$

Subtract from both sides:

$(2 x + 10) - 10 = - 3 - 10$

Simplify the arithmetic:

$2 x = - 3 - 10$

Simplify the arithmetic:

$2 x = - 13$

Divide both sides by :

$\frac{(2 x)}{2} = \frac{- 13}{2}$

Simplify the fraction:

$x = \frac{- 13}{2}$

10 additional steps

$(6 x + 10) = - (4 x - 3)$

Expand the parentheses:

$(6 x + 10) = - 4 x + 3$

Add to both sides:

$(6 x + 10) + 4 x = (- 4 x + 3) + 4 x$

Group like terms:

$(6 x + 4 x) + 10 = (- 4 x + 3) + 4 x$

Simplify the arithmetic:

$10 x + 10 = (- 4 x + 3) + 4 x$

Group like terms:

$10 x + 10 = (- 4 x + 4 x) + 3$

Simplify the arithmetic:

$10 x + 10 = 3$

Subtract from both sides:

$(10 x + 10) - 10 = 3 - 10$

Simplify the arithmetic:

$10 x = 3 - 10$

Simplify the arithmetic:

$10 x = - 7$

Divide both sides by :

$\frac{(10 x)}{10} = \frac{- 7}{10}$

Simplify the fraction:

$x = \frac{- 7}{10}$

3. List the solutions

$x = - \frac{13}{2}, - \frac{7}{10}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 6 x + 10 |$
$y = | 4 x - 3 |$
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 x

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 x

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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