Can someone please help me with these questions?

Question

Solve each absolute value inequality and graph the solution set. I realize the graphing might be hard to do. Whatever help I can get will be great!

l x-7l<= -4


1>1/2 l6-xl -3/4


x+8y>8 and
x-2y<10



lx-3yl >=9

Answer 1

A key thing to know is how absolute values work with inequalities. Here's an example.

|y| < 7

Means

-7 < y < 7 (i.e. the value of y lies in between -7 and 7)

Also,

|y| > 7 means

y > 7 or y < -7

That's all you have to keep in mind. Let's give these questions a shot.

|x - 7| <= -4

This is a FALSE statement, because the absolute value is ALWAYS greater than or equal to 0.


For

1 > 1/2|6-x| - 3/4

Solve the inequality by isolating the absolute value. Add 3/4 to both sides,

1 + 3/4 > 1/2 |6 - x|

Solve the left hand side using a common denominator,

7/4 > 1/2 |6 - x|

Multiply both sides by 4

7 > 2 | 6 - x |

And then divide both sides by 2

7/2 > | 6 - x |

We can just flip these terms as long as we flip the sign.

|6 - x| < 7/2

Therefore, this translates to

-7/2 < 6 - x < 7/2

Subtracting everything by 6, we get

-7/2 - 6 < -x < 7/2 - 6

or

-19/2 < -x < -5/2

Multiply everything by -1 would essentially flip all signs, since multiplying or dividing by a negative number does this with inequalities.

19/2 > x > 5/2

And that's our answer, which is saying x lies strictly between 19/2 and 5/2.