How to solve this absolute value inequality?

If this was an equation, removing the absolute value places a ± on the other side.

Since this is an inequality, the - side of that has to have the sign inverted, so we have:

|2x + 4| < x

2x + 4 < x and 2x + 4 > -x

solve for each:

x < -4 and 3x > -4

x < -4 and x > -4/3

There are no numbers less than -4 and greater than -4/3, so there are no solutions to this equality.

This can't really be done algebraically since absolute values do not have an operational "opposite". We can, however look at how these functions will behave seperately.

Think about what it would look like if you were to set both sides equal to y and graph them as two seperate equations. The LHS would touch the x-axis at -4 and shoots up with a slope of 2 and -2 in the opposite direction (both shoot up from the point touching the x-axis). The RHS is just a simple y=x so it touches the x-axis at 0 and shoots up with a slope of 1. Now, since the LHS is an absolute function, all of its other points lie above the x-axis, and the y=x function does not have a steep enough slope to ever reach the other function, so the two functions do not touch. This means the inequality given is never true at any point. THis could also be verified by solving this set of equations:

y=x

y=|2+4|

You would find that these do not have a solution.

Graph those two functions if you want to visually see what I meant earlier.

There are no real solutions.

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The least value on the left is where x=-2. At that point, the expression evaluates to

.. 0 < -2 ... FALSE

To the left of that point, the left side expression increases while the right side expression decreases. To the right of that point, the left side expression increases faster than the right side expression.