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Solution - Absolute value equations

Exact form:

x = \frac{7}{6}, - \frac{7}{4}

x=\frac{7}{6} , -\frac{7}{4}

Mixed number form:

x = 1 \frac{1}{6}, - 1 \frac{3}{4}

x=1\frac{1}{6} , -1\frac{3}{4}

Decimal form:

x = 1.167, - 1.75

x=1.167 , -1.75

See steps

Step-by-step explanation

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

$| x - 7 | + | 5 x | = 0$

Add $- | 5 x |$ to both sides of the equation:

$| x - 7 | + | 5 x | - | 5 x | = - | 5 x |$

Simplify the arithmetic

$| x - 7 | = - | 5 x |$

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
$| x - 7 | = - | 5 x |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| x - 7 \| = - \| 5 x \|$
$x = + y$	$(x - 7) = - (5 x)$
$x = - y$	$(x - 7) = - - (5 x)$
$+ x = y$	$(x - 7) = - (5 x)$
$- x = y$	$- (x - 7) = - (5 x)$

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:

$\| x \| = \| y \|$	$\| x - 7 \| = - \| 5 x \|$
$x = + y$ , $+ x = y$	$(x - 7) = - (5 x)$
$x = - y$ , $- x = y$	$(x - 7) = - - (5 x)$

3. Solve the two equations for x

7 additional steps

$(x - 7) = - 5 x$

Add to both sides:

$(x - 7) + 7 = (- 5 x) + 7$

Simplify the arithmetic:

$x = (- 5 x) + 7$

Add to both sides:

$x + 5 x = ((- 5 x) + 7) + 5 x$

Simplify the arithmetic:

$6 x = ((- 5 x) + 7) + 5 x$

Group like terms:

$6 x = (- 5 x + 5 x) + 7$

Simplify the arithmetic:

$6 x = 7$

Divide both sides by :

$\frac{(6 x)}{6} = \frac{7}{6}$

Simplify the fraction:

$x = \frac{7}{6}$

12 additional steps

$(x - 7) = - - 5 x$

Group like terms:

$(x - 7) = (- 1 \cdot - 5) x$

Multiply the coefficients:

$(x - 7) = 5 x$

Subtract from both sides:

$(x - 7) - 5 x = (5 x) - 5 x$

Group like terms:

$(x - 5 x) - 7 = (5 x) - 5 x$

Simplify the arithmetic:

$- 4 x - 7 = (5 x) - 5 x$

Simplify the arithmetic:

$- 4 x - 7 = 0$

Add to both sides:

$(- 4 x - 7) + 7 = 0 + 7$

Simplify the arithmetic:

$- 4 x = 0 + 7$

Simplify the arithmetic:

$- 4 x = 7$

Divide both sides by :

$\frac{(- 4 x)}{- 4} = \frac{7}{- 4}$

Cancel out the negatives:

$\frac{4 x}{4} = \frac{7}{- 4}$

Simplify the fraction:

$x = \frac{7}{- 4}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 7}{4}$

4. List the solutions

$x = \frac{7}{6}, - \frac{7}{4}$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | x - 7 |$
$y = - | 5 x |$
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 x

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 x

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved