The easy part is isolating the absolute-value term:
5 + 7 |2<em>x</em> - 1| = -44
7 |2<em>x</em> - 1| = -49
|2<em>x</em> - 1| = -7
Remember that the absolute value function returns a positive number that you can think of as the "size" of that number, or the positive distance between that number and zero. If <em>x</em> is a positive number, its absolute value is the same number, |<em>x</em>| = <em>x</em>. But if <em>x</em> is negative, then the absolute value returns its negative, |<em>x</em>| = -<em>x</em>, which makes it positive. (If <em>x</em> = 0, you can use either result, since -0 is still 0.)
The important thing to take from this is that there are 2 cases to consider: is the expression in the absolute value positive, or is it negative?
• If 2<em>x</em> - 1 > 0, then |2<em>x</em> - 1| = 2<em>x</em> - 1. Then the equation becomes
2<em>x</em> - 1 = -7
2<em>x</em> = -6
<em>x</em> = -3
• If 2<em>x</em> - 1 < 0, then |2<em>x</em> - 1| = - (2<em>x</em> - 1) = 1 - 2<em>x</em>. Then
1 - 2<em>x</em> = -7
-2<em>x</em> = -8
<em>x</em> = 4
