聯立不等式 3 (英)
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聯立不等式 3 (英): 聯立不等式 3
- We're asked to solve for x and we have this compound inequality here. Negative sixteen is less than or equal
- to three x plus five which is less than or equal to twenty. And really there's two ways to approach it which are
- really the same way and I'll do both of them. And I'll actually do both of them simultaneously. So
- one is to just solve this compound inequality all at once. I'll just rewrite it. Negative sixteen
- is less than or equal to three x plus five which is less than or equal to twenty. And the other way
- is to think of it as two separate inequalities but both of them need to be true. So you could also view
- it as negative sixteen has to be less than or equal to three x plus five, and three x plus five needs
- to be less than or equal to twenty. This statement and this statement are equivalent. This one may
- seem a little bit more familiar because we can independently solve each of these inequalities and just
- remember the AND. This one might seem a little less traditional because now we have three sides to
- the statement. We have three parts of this compound inequality. But we can see that we are actually going
- to solve it the exact same way. In any situation we really just want to isolate the x on one side of the inequality,
- or in this case one part of the compound inequality. Well the best way to isolate this x right here
- is to first get rid of this positive five that is sitting in the middle. So lets subtract five from
- every part of this compound inequality. Subtract five there, subtract five there, and subtract five over there.
- So we get, negative sixteen minus five is negative twenty-one, is less than or equal to 3x plus
- five minus five is 3x, which is less than or equal to twenty minus five which is fifteen. And we can essentially
- do the same thing here. If we want to isolate the 3x, we can subtract five from both sides, subtract
- five from both sides, we get negative twenty-one. Negative twenty-one is less than or equal to 3x.
- And we get, subtracting five from both sides, and notice we are just subtracting five from every part of this compound inequality.
- We get 3x is less than or equal to fifteen. So this statement and this statement are once again the
- exact same thing. Now going back here if we want to isolate the x, we divide by three. We have to do
- it to every part of the inequality. And since three is positive we don't have to change the sign.
- So let's divide every part of this compound inequality by three. This is equivalent to dividing every
- part of each of these inequalities by three. And then we get, negative twenty-one divided by three is negative
- seven, is less than or equal to x, which is less than or equal to fifteen divided by three is five.
- You do it over here, you get negative seven is less than or equal to x, and x is less than or equal to
- fifteen over three which is five. This statement and this statement are completely equal. Now we've
- solved for x. We've given you the solution set. And if we want to graph it on a number line it would
- look like this. This is zero, this is five, this is negative seven. Our solution set includes everything
- between negative seven and five, including negative seven and five. So we have to fill in the circles
- on negative seven and positive five, and it is everything in between. That's our solution set.
- And so we can verify that these work. You can try out a number that's well inside
- of our solution set like zero. Three times zero is zero. So you're just left with five is greater
- than or equal to negative sixteen, which is true. And five is less than or equal to twenty.
- Or negative sixteen is less than or equal to five, which is less than or equal to twenty.
- So that works, and that makes sense. You could try five. If you put five here you get three times five
- plus five, well that's just twenty. Negative sixteen is less than or equal to twenty, which is less
- than or equal to twenty. That works. Negative seven should also work.
- Three times negative seven is negative twenty-one plus five is negative sixteen. So you get negative
- sixteen, which is less than or equal to negative sixteen, which is less than or equal to twenty.
- And you could try other values. You could go outside of our solution set. Try something like ten.
- Ten should not work. If you put ten here, you get three times ten plus five is thirty-five.
- Negative sixteen is less than or equal to thirty-five, but thirty-five is not less than or equal to twenty.
- And that's why ten is not part of our solution set.