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聯立不等式 3 (英)

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聯立不等式 3 (英) : 聯立不等式 3

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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.

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聯立不等式 3 (英)

聯立不等式 3 (英) : 聯立不等式 3