Solve Linear Inequalities
So when we’re solving an inequality
such as this one, we want to pretend that the inequality isn’t there; it’s just like
a normal equal sign. And then we can solve it as we
would any normal linear equation, so we will solve this inequality as if it was
equal to two 𝑥 minus three equal seven. So if we had two 𝑥 minus three
equal seven, we could see it’s just a simple two-step equation. So the first thing we would do is
get rid of the minus three by adding three to both sides.
And then on the left-hand side,
that gives us two 𝑥; and on the right-hand side, we have seven plus three, which is
ten. We cannot forget to put the sign in
exactly as it is.
Don’t swap it halfway through
because otherwise it will change the value of what you’re saying. Now we’ve got two multiplied by
𝑥. The opposite of multiply is
divide. Whatever we do to one side we have
to do to the other, so we’re going to have to divide both sides by two.
And now on the left-hand side, that
just gives us 𝑥, which is greater than ten divided by two, which is five. And there we have it, we have
solved that inequality for 𝑥. So we are saying the original
linear inequality is the same as saying 𝑥 is greater than five.
Now looking at this next question
we’ve got six minus two 𝑥 all divided by three is less than six. So first thing we’ll need to do is
get rid of that divide by three because we can’t do anything else. So the opposite of divide by three
we know it’s multiply by three, so we’re going to multiply both sides by three. And that gives us six minus two 𝑥
is less than six times by three, which is eighteen.
And then it looks like a linear
inequality now, so we can see what we could do if we wanted is add two 𝑥 to both
sides and then subtract eighteen from both sides. But what we’re gonna do instead
this time is subtract six from both sides. And that will give us negative two
𝑥 is less than twelve.
And because we have a negative, we
have to if we multiply or divide an inequality by a negative, we have to swap the
inequality. So if it’s less than, it has to
become greater than. So we could see we’ve got negative
two multiplied by 𝑥. So to get rid of a negative two
multiplied by 𝑥, we have to divide by negative two.
So we know negative two 𝑥 divided
by negative two is just 𝑥, and twelve divided by negative two is negative six. And then as I said, when we
multiply it or divide by a negative, we must swap the sign around. So we can see right now it’s less
than, so it needs to become greater than.
So if you want to see why we have
to swap the inequality around, let’s have a look about what would have happened if
instead of just dividing by negative two, we had moved the negative two 𝑥 onto the
right-hand side and the twelve onto the left-hand side.
So we would’ve added two 𝑥 to both
sides, which would’ve given us zero is less than twelve plus two 𝑥. And then we would have subtracted
twelve from both sides, which would have given us negative twelve is less than two
𝑥. And finally, we would’ve divided
both sides by two, giving us negative six is less than 𝑥, which we can see is
exactly the same as 𝑥 is greater than negative six. So we’ve saved ourselves all of
that hassle of moving the 𝑥s and moving the numbers onto different sides by just
being able to divide by a negative and swapping the inequality.