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LOTS OF POINTS!!!!!!!! HELP PLEASE RSM IS DUE TOMORROW!!!!!!!

For which values of 'a' does the system have at least one solution?

LOTS OF POINTS!!!!!!!! HELP PLEASE RSM IS DUE TOMORROW!!!!!!! For which values of-example-1
User Karan Dua
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1 Answer

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19 votes

Answer: Anything larger than the fraction 21/127

21/127 = 0.16535 approximately

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Step-by-step explanation:

Let's solve each inequality for x

We'll start with the first inequality


3(a-5x) < 1+x\\\\3a-15x < 1+x\\\\3a-1 < x+15x\\\\3a-1 < 16x\\\\16x > 3a-1\\\\x > (3a-1)/(16)

The variable x is larger than (3a-1)/16. If we knew what the value of 'a' was, then we could nail down the range of values for x more concretely.

Do the same for the second inequality


2 - (x)/(2) > 3 + 5(x-a)\\\\2 - (1)/(2)x > 3 + 5(x-a)\\\\2 - 0.5x > 3 + 5(x-a)\\\\2 - 0.5x > 3 + 5x-5a\\\\2 + 5a-3 > 5x+0.5x\\\\5a-1 > 5.5x\\\\5.5x < 5a-1\\\\x < (5a-1)/(5.5)\\\\

The variable x is also less than (5a-1)/(5.5)

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At this point, we have found that


x > (3a-1)/(16) and
x < (5a-1)/(5.5)

This means x is between those endpoints, excluding each endpoint

So we can form this compound inequality


(3a-1)/(16) < x < (5a-1)/(5.5)

This tells us the range of where x spans from.

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If the endpoints are equal, aka the same value, then there's no way to have x have any solutions

So let's equate the endpoints and see what happens


(3a-1)/(16) = (5a-1)/(5.5)\\\\5.5(3a-1) = 16(5a-1)\\\\16.5a - 5.5 = 80a - 16\\\\-5.5 + 16 = 80a - 16.5a\\\\10.5 = 63.5a\\\\63.5a = 10.5\\\\a = (10.5)/(63.5)\\\\a = (105)/(635)\\\\a = (21*5)/(127*5)\\\\a = (21)/(127)\\\\a \approx 0.16535\\\\

If 'a' is equal to that value, then the two endpoints of that compound inequality are completely identical; therefore, in that situation, we wouldn't have any solutions for x.

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The question is: what happens if 'a' is some number smaller than 21/127?

Let's say a = 0

  • The left endpoint would be (3a-1)/16 = (3*0-1)/16 = -0.0625
  • The right endpoint would be (5a-1)/(5.5) = (5*0-1)/(5.5) = -0.1818 approximately

So we see that -0.0625 < x < -0.1818; however, upon closer inspection, you should find that such an interval makes no sense. Why not? Because -0.0625 is not smaller than -0.1818. It should be the other way around. No number exists that is between -0.0625 and -0.1818 (I recommend using a number line to help see why this is the case).

Therefore, anything smaller than a = 21/127 will result in having no solutions for x in the original system of inequalities.

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Now let's try something larger than a = 21/127

Let's say we picked a = 1

  • The left endpoint would be (3a-1)/16 = (3*1-1)/16 = 0.125
  • The right endpoint would be (5a-1)/(5.5) = (5*1-1)/(5.5) = 0.7273 approximately

We end up with 0.125 < x < 0.7273 which is now valid because 0.125 is indeed smaller than 0.7273

Therefore, if a > 21/127, then we'll have those original system of inequalities lead to more than one solution for x.

User Hkutluay
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