SOLUTION: Solve the inequality |x - 2| <= |3x + 12| + 2.

Question 1183626: Solve the inequality
|x - 2| <= |3x + 12| + 2.
Found 2 solutions by MathLover1, greenestamps:
Answer by MathLover1(20850) (Show Source): You can put this solution on YOUR website!

Let's solve your inequality step-by-step.

Let's find the critical points of the inequality.

|

Either or

Part 1:
............(Flip the equation)
...........(Add to both sides)

We know either or

...............(Possibility )
........... (Subtract from both sides)

............ (Subtract from both sides)

............ (Divide both sides by )

........... (Possibility )
............ (Simplify both sides of the equation)
................ (Add to both sides)

.................... (Subtract from both sides)

........ (Divide both sides by )

Check answers. (Plug them in to make sure they work.)
=>Doesn't work in original equation
=>Doesn't work in original equation

Part 2:
............(Flip the equation)

......... (Add to both sides)

.............(Divide both sides by)

We know either or

.......... (Add to both sides)

........... (Subtract from both sides)

......... (Divide both sides by )

............. (Possibility 2)
............ (Simplify both sides of the equation)
.................... (Subtract from both sides)

.............(Subtract from both sides)

........... (Divide both sides by)

Check answers. (Plug them in to make sure they work.)
=> Works in original equation.
=>Works in original equation.

Critical points:
or
Check intervals in between critical points. (Test values in the intervals to see if they work.)
=>Works in original inequality.
=> Doesn't work in original inequality.
=>Works in original inequality.
Answer:
or

Answer by greenestamps(13209) (Show Source): You can put this solution on YOUR website!

Probably the easiest way to find the solution set is to graph the two expressions with a graphing calculator.

red:
green:

We can see from the graph that the solution set (red line below green line) is (-infinity, -6] U [-3,infinity).

For an algebraic solution, I prefer a different method than the one shown by the other tutor....

The behaviors of the two functions change when (x-2) is zero and when (3x+12) is zero -- at x = -4 and x=2. So separate the analysis into three intervals:
(1) (-infinity,-4]
(2) [-4,2]
(3) [2,infinity)

(1) On (-infinity,-4]...

All the x values in that range are in the interval for this case, so part of the solution set is (-infinity,-6].

(2) On [-4,2]...

For x values in the interval for this case, [-4,2], only the values x>=-3 satisfy the inequality. So another part of the solution set is [-3,2].

(3) On [2,infinity)...

All the x values in that range are in the interval for this case, so part of the solution set is [2,infinity).

Combining the solutions sets for the three cases we get the answer we saw from the graph: (-infinity,-6] U [-3,infinity)

RELATED QUESTIONS
Solve the inequality -2x - 12 > -3x - 2 for... (answered by Fombitz,Edwin McCravy)
solve the inequality... (answered by richwmiller)
Solve the following Compound Inequality. 4x<2 x-10 or... (answered by josgarithmetic)
Solve the inequality:... (answered by tommyt3rd)
solve the inequality 2(x + 6) >= 3x -... (answered by jim_thompson5910)
Solve the inequality... (answered by Edwin McCravy)
solve the linear inequality... (answered by waynest)
solve the inequality. THANKS!... (answered by praseenakos@yahoo.com)
Solve the inequality. {{{ -3x-2<_-5... (answered by Fombitz)