One other type of absolute value problem is when two absolute values are equal to each other. We still will consider both the positive and negative result, the difference here will be that we will have to distribute a negative into the second absolute value for the negative possibility. Example `|2x − 7| = |4x + 6|`

`|2x − 7| = |4x + 6|`	Absolute value can be positive or negative
`2x−7=4x+6 or 2x−7=−(4x+6)`	make second part of second equation negative

`2x−7=4x+6`	`2x−7=-4x-6`
`2xcolor(red)(-2x)−7=4xcolor(red)(-2x)+6`	`2xcolor(red)(4x)−7=-4xcolor(red)(4x)-6`
`-7 = 2x+6 `	`6x-7=-6`
`-7color(red)(-6)= 2x+6color(red)(-6)`	`6x-7color(red)(+7)=-6color(red)(+7)`
`-13=2x`	`6x = 1`
`(-13)/color(red)(2)=(2x)/color(red)(2) `	`(6x)/color(red)(6) = (1)/color(red)(6)`
`-13/2=x `	`x=1/6`

This gives us our two solutions, `x= −13/2` or `x= 1/6`. As we are solving absolute value equations it is important to be aware of special cases. Remember the result of an absolute value must always be positive. Notice what happens in the next example. Example `7 + |2x − 5| = 4`

`7 + |2x − 5| = 4 `	Notice absolute value is not alone
`7color(red)(-7) + |2x − 5| = 4color(red)(-7)`	Subtract 7 from both sides
`|2x − 5| = − 3`	Result of absolute value is negative!

Notice the absolute value equals a negative number! This is impossible with abso- lute value. When this occurs we say there is no solution or ∅.