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- Solve for x, 5x - 3 is less than 12 "and" 4x plus 1 is greater than 25.
- So let's just solve for X in each of these constraints and keep in mind
- that any x has to satisfy both of them
- because it's an "and" over here
- so first we have this 5 x
- minus 3 is less than 12
- so if we want to isolate the x
- we can get rid of this negative 3
- here by adding 3 to both sides
- so let's add 3 to both sides of this inequality.
- The left-hand side, we're just left with a 5x, the minus 3 and the plus 3 cancel out.
- 5x is less than 12 plus 3 is 15.
- Now we can divide both sides by positive 5,
- that won't swap the inequality since 5 is positive.
- So we divide both sides by positive 5
- and we are left with just from this constraint
- that x is less than 15 over 5, which is 3.
- So that constraint over here.
- But we have the second constraint as well.
- We have this one, we have 4x plus 1
- is greater than 25.
- So very similarly we can substract one from both sides
- to get rid of that one on the left-hand side.
- And we get 4x, the ones cancel out.
- is greater than 25 minus one is 24.
- Divide both sides by positive 4
- Don't have to do anything to the inequality since
- it's a positive number.
- And we get x is greater than
- 24 over 4 is 6.
- And remember there was that "and" over here.
- We have this "and".
- So x has to be less than 3 "and" x has to be greater than 6.
- So already your brain might be realizing
- that this is a little bit strange.
- This first constraint says that x needs to be less than 3
- so this is 3 on the number line.
- We're saying x has to be less than 3
- so it has to be in this shaded area right over there.
- This second constraint says that x has to be
- greater than 6.
- So if this is 6 over here, it says that x has to greater than 6.
- It can't even include 6.
- And since we have this "and" here.
- The only x-es that are a solution for this compound inequality
- are the ones that satisfy both.
- The ones that are in the overlap
- of their solution set.
- But when you look at it right over here it's clear that
- there is no overlap.
- There is no x that is both greater than 6 "and" less than 3.
- So in this situation we no solution.

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