How the do you solve this inequality |1/2x+2|<4 (The < is less than or equal to)?

Question

This is crazy, I never seen this before or was taught it.

Answer 1

In an absolute value inequality, < is an "and" situation and > is an "or" situation. In either case, you have to do it twice, once for the positive answer and once for the negative answer. This is because if the absolute value of x is 5, x could be 5 or -5. Also, the direction of the sign changes for the negative value.

1/2 x + 2 < 4 and 1/2 x + 2 > -4
1/2 x < 2
x < 4

1/2 x + 2 > -4
1/2 x > -6
x > -12

So x < 4 and x > -12

Answer 2

Since it is an absolute value, what's in the | | should be set to + and - the other side of the equation.
So you have:
1/2 x +2 <=4 and 1/2 x + 2 <= -4

Solve both for your x values.
a) x <= 4 b) x <=-12

Since these overlap, you really just have x <= 4.

P.S. don't just do what KC did because the answers will not usually overlap and you will miss half of the answer!

Answer 3

x>1/4 and -1/12

Answer 4

Is it ½x (which is x/2)? or is it 1/(2x)? I am going to assume the first.

|x/2 + 2| <= 4

Can be rewritten as:

-4 <= x/2 + 2 <= 4

Subtract 2 from all sides, and you get:

-6 <= x/2 <= 2

Multiply all sides by 2, and you get:

-12 <= x <= 4

And there you have it. :-)

Answer 5

X is less than 4.
Subtract 2 from eah side, then multiply each side by 2.

(1/2X+2)<4
1/2X<2
X<4

Answer 6

a million) 2x+5_<11 and four(x-a million)+12_>0 2x _< 11 - 5 and 4x - 4 + 12 _> 0 2x_< 6 and 4x _> -8 x _< 6/2 and x_> 8/4 x_<3 and x_>2 answer: {x belongs to R | 2<_x_<3} 2) 5-2x>-3 or 3(6-x)<2(x-6) -2x > -3-5 or 18 - 3x < 2x - 12 -2x > -8 or -3x - 2x < -12 - 18 x <-8/-2 or -5x < -30 x < 4 or x > -30/-5 x < 4 or x > 6 answer: {x factors of R| 4>x>6} 3) 3x>4-x or 2(x+3)>5x-3 3x + x > 4 or 2x + 6 > 5x -3 4x > 4 or 2x - 5x > -3 -6 x > 4/4 or -3x > -9 x > a million or x <-9/-3 x > a million or x < 3 answer: {x belongs to R | a million< x < 3} ><