Question:

X - 2 > 2x + 1 or -10 > -2x - 2 ?

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Here is a sample "word" description:

The solution is x > 5 and x < 9. There would be a open circle on the 5 and a closed circle on the 9. The shading is between the two numbers.

Can You Help Me Figure these Out "Word" description?

Thanks so much!!

2x > -6 and x - 4 <= 3

x + 5 >= 2x + 1 and -4x < -8

-6 < x + 3 < 6 (Remember this is an "and" statement)

-3x < -6 or x + 5 <= -2

x - 2 >= 2x + 1 or -10 > -2x - 2

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Do you need help writing word descriptions for each of these?  Is that what you are asking?

If you are then first solve for x in each equation, and then write a word description.

For 2x > -6 and x - 4 <= 3  solve first for x on each side.

divide the 2x > -6 by 2, and add four to both sides of x - 4 <= 3.

2x/2 > -6/2 and x - 4 + 4<= 3 + 4

Therefore: x > -3 and x < or = 7.  There would be an open circle on x = -3 and a closed circle on x = 7, with a line connecting the two.

Does this make sense?  See, you place closed circles on numbers that are = a value (they can be <= or >=).  The open circles imply that the value is not included, so in this last example, when x > -3, it equaled every value even -2.9999999999999999999999, but could not equal -3.

For x + 5 >= 2x + 1 and -4x < -8 solve for x just like last time.

To do this, just focus on one statement at a time.

Start with the statement to the left of the "and" x + 5 >= 2x + 1

You are going to want to combine like terms (put all the x's to one side and the numbers to another).

To do this first subtract 1 from both sides (this will eliminate a number value from the right side of the equation).

x + 5  - 1 >= 2x + 1 - 1

x + 4 >= 2x

Now, subtract x from both sides (this will put the x's to one side and the numbers to another).

x + 4 - x >= 2x - x

4 >= x

Now, look at the right side of your problem x + 5 >= 2x + 1 and -4x < -8.  (The part after the "and")

-4x < -8 To get x by itself you need to divide both sides by - 4.

-4x/-4 < -8/-4

x > 2  *Remember that when you divide an inequality (< or >) by a negative value, the sign changes.

Now you know that x + 5 >= 2x + 1 and -4x < -8 solves out to:

x<= 4 and x>2.  There would be an open circle on the 2 (or x = 2) and a closed circle on 4 (or x = 4), with a line connecting the two circles.

For -6 < x + 3 < 6 you can change this into something that looks a little more familiar, by using the hint that was given:  (Remember this is an "and" statement).

-6 < x + 3 < 6 can be changed into -6 < x + 3  and  x + 3 < 6

Now, break the statement apart again, by starting with the equation to the left of the "and".

-6 < x + 3.  Solve this for x

Subtract 3 from both sides.

-6 - 3 < x + 3 - 3

-9 < x

Now look at the equation to the right of the "and".

x + 3 < 6.  Solve this for x

Subtract 3 from both sides.

x + 3 - 3 < 6 - 3

x < 3

Now you know that -6 < x + 3 < 6 solves out to:

x > -9 and x < 3.  There will be no closed circles (because there are no = signs) only open circles.  One open circle will be on -9 (or x = -9) and the other will be on 3 (or x = 3).  A line will connect the two open circles.

Although "or" statements function a little differently, it is really only in their word descriptions that they deviate from "and" ones.  Thus, they should be solved the same way.

For -3x < -6 or x + 5 <= -2 break the inequality statement apart by taking the inequality to the left of the "or".

-3x < -6.  Solve this for x.

Divide both sides by -3.

-3x/-3 < -6/-3

x > 2 (Again, remember because you are dividing the inequality by a negative the sign must change from < to >).

Now look at the right side of the "or".

x + 5 <= -2.  Solve for x.

Subtract 5 from both sides.

x + 5 - 5 <= -2 - 5

x <= -7

Therefore the solution to -3x < -6 or x + 5 <= -2 is:

x > 2 or x <= -7.  There will be a closed circle on -7 (or x = -7) and an open circle on 2 (or x = 2).  There will be a line drawn from the closed circle on -7 backwards toward -8, -9, -10 and so on, and a line drawn forward from the open circle toward 3, 4, 5, etc.

This is because it is an or statement.  The inequality is telling you, I am either -7 or a number less than that, or I am a number greater that 2 and nothing in between.  That is why the lines are drawn away from the circles, and not connecting the two.

Finally, for x - 2 >= 2x + 1 or -10 > -2x - 2, break the statement apart as you have done for the last few problems.

Look as the inequality to the left of the "or".

x - 2 >= 2x + 1.  Combine like terms, starting with your normal numbers.

Subtract 1 from both sides.

x - 2 - 1 >= 2x + 1 - 1

x - 3 >= 2x.

Now subtract x from both sides.

x - 3 - x >= 2x - x

-3 >= x

Look at the inequality to the right of the "or".

-10 > -2x - 2.  Solve for x.

Add 2 to both sides.

-10 +2 > -2x - 2 + 2

-8 > -2x

Divide both sides by -2

-8/-2 > -2x/-2

4 < x

Therefore you found that x - 2 >= 2x + 1 or -10 > -2x - 2 solves to:

x>4 or x <= -3.  There will be a closed circle on -3 (or x = -3) and an open circle on 4 (or x = 4).  There will be a line leading away from the closed circle, going toward -4, -5, etc., and a line leading away from the open circle going toward 5, 6, etc.

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