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Question 561067: HELP PLEASE!!!!! solve the compound inequality 7 > -3x +6 or 10 <= -2x + 5
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! your equation is:
7 > -3x + 6 or 10 <= -2x + 5
solve both equations separately.
first equation is:
7 > -3x + 6
add 3x to both sides of this equation and subtract 7 from both sides of this equation to get:
3x > 6 - 7
simplify to get:
3x > -1
divide both sides of this equation by 3 to get:
x > -1/3 *****
second equation is:
10 <= -2x + 5
add 2x to both sides of this equation and subtract 10 from both sides of this equation to get:
2x <= 5 - 10
simplify to get:
2x <= -5
divide both sides of this equation by 2 to get:
x <= -5/2 *****
you have:
x > -1/3 or x <= -5/2
the value of x will satisfy the first equation or it will satisfy the second equation but it cannot satisfy both equations at the same time because there is no overlap between the value of x that will satisfy the first equation and the value of x that will satisfy the second equation.
there is a dead space between them that will satisfy neither equation.
that dead space is between x > -5/2 and x <= -1/3
when x falls in that space on the number line, you will see that neither equation is satisfied.
the following table will help to illustrate this:
x e1 e2 7>e1 10<=e2 limits
3/4 3.75 3.50 yes ---
1/5 5.38 4.58 yes ---
- 1/3 7.00 5.67 --- --- x = -1/3
- 7/8 8.63 6.75 --- ---
-1 3/7 10.25 7.83 --- ---
-2 11.88 8.92 --- ---
-2 1/2 13.50 10.00 --- yes x = -5/2
-3 15.13 11.08 --- yes
the value of x is going from more positive to more negative (larger value to smaller value).
e1 is the first equation of -3x+6
e2 is the second equation of -2x+5
you can see that 7 is greater than e1 when x is greater than -1/3
you can see that 10 is smaller than or equal to e2 when x is smaller than or equal to -2 and 1/2 which is the same as -5/2.
this agrees with our conclusion that:
x > -1/3 or x <= -5/2
you can also see that there is a dead space between x = -1/3 and x < -5/2 where neither equation is satisfied.
for example:
when x = -2, e1 = 11.58 and e2 = 8.92
7 is not greater than e1
10 is not smaller than or equal to e2.
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