Bow tie inequality: graphing | y | <= | x |

[jeennyjeon]

hi i'm in algebra II, and this challenge problem is very~ confusing..

Graph the bowtie inequality, (system of inequalities)
|y| ? |x|.

so, i thought this meant y ? |x|, and x ? |y|.
i got the y is less than or equal to |x|, but x ? |y| is the confusing part..
because that means x can't be negative..

and, to get to the bowtie inequality, (i tried working backwards)
it's y ? |x| and y ? -|x|.

but this does make sense, but then i came up with another question...
if y ? -|x| is true, then -y ? |x| must be true also, which doesn't make sense because y can't be negative because absolute values can't be negative...

i'm SOO confused~~ HELP MEEE!

 

[mmm4444bot]

Re: Bow tie inequality

Graph the bowtie inequality, (system of inequalities)
|y| ? |x|.

so, i thought this meant y ? |x|, It does. This applies in Quadrant II.

and x ? |y|. This applies in Quadrant IV.

if y ? -|x| is true, This is not correct.

then -y ? |x| must be true also, Did you multiply by -1 to get this? If so, then you forgot to reverse the inequality symbol.

which doesn't make sense because y can't be negative -y is not always negative ...

Hi Jeenny:

Try starting with the equality first, since that is how we determine the boundary of the shaded region for the graph of an inequality.

|y| = |x|

Removing the absolute value symbols gives us two equations.

Do you know what they are?

Graph these two lines.

When considering which region to shade, consider the following.

Any point on the x-axis has a y-coordinate of zero. Clearly, zero is less than or equal to the absolute value of all real numbers. Therefore, no matter what the value of x is, any point on the x-axis will satisfy the original inequality.

In other words, the entire x-axis is part of the solution, so it must lie somewhere within the shaded region.

Also, you can make a pre-sketch, and play around with points in each quadrant, comparing the x-coordinate with the y-coordinate. If you play around for awhile, the meaning of the bowtie system of inequalities may become apparent to you.

If it does not become apparent, then test an arbitrary point in each quadrant so see which side of the line to shade in that quadrant.

Let us know if you need more help with this exercise. Thank you for sharing your work.

~ Mark

 

[jeennyjeon]

okay. so i think i got it. the shades of the inequalities are the UPPER PORTION and the BOTTOM PORTION, making more of an hourglass.
but the white space makes a bowtie.
did i do it right?

 

[mmm4444bot]

... the shades of the inequalities are ... more of an hourglass ... did i do it right?

I don't think so.

Pick any point in the shaded region. Substitute the coordinates of this test point into the original inequality statement.

Is it true? I mean, do those particular values for x and y make the given inequality a true statement?

~ Mark