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.