Video: Solving System of Linear Inequalities in One Variable

Solve 10𝑥 + 1 < 𝑥 − 8 and 𝑥 − 1 > −5.

04:16

Video Transcript

Solve 10𝑥 plus one is less than 𝑥 minus eight and 𝑥 minus one is greater than negative five.

So in this question, what we’re gonna do is actually solve each of the inequalities separately and then bring the solutions together. And just to actually help us understand the question and what we’re working with, I’m just gonna quickly recap bit of the inequality notation that we’re using.

So if you’ve got 𝑎 and then you’ve got the wide side of our inequality sign 𝑏, this means that 𝑎 is greater than 𝑏. If you’ve got 𝑎 by the pointy end of our inequality sign and then 𝑏 at the wider end, we can say that 𝑎 is less than 𝑏.

Okay, we’re gonna start by solving 10𝑥 plus one is less than 𝑥 minus eight. So when we’re actually gonna solve an inequality, we solve it in much the same way that we’d solve an equation.

So therefore, the first step with this inequality is to actually subtract one from each side of our inequality. And when we do that, we get 10𝑥 — that’s because we had 10𝑥 plus one and we subtracted one, so it leaves us with 10𝑥 — is less than then we’ve got 𝑥 minus nine — that’s cause if you have negative eight and you subtract one, it’s like going left on our number line, so we go to negative nine.

And now as we only want letters on one side of our inequality and numbers on the other, what we’re gonna do is subtract 𝑥 from each side of our inequality. And when we do that, we get nine 𝑥, because 10𝑥 minus 𝑥 is nine 𝑥, is less than negative nine. So then, finally, what we’re gonna do is actually divide each side of our inequality by nine. And when we do that, we’re left with 𝑥 is less than negative one. So that’s the solution to our first inequality.

So now let’s move on to our second inequality. So the second inequality is 𝑥 minus one is greater than negative five. So what we’re gonna do here is actually add one to each side of the inequality. And when we do that, we’re gonna get 𝑥 is greater than negative four. So now that’s our second inequality solved.

So what we need to do now is bring our solutions together, because what we want to do is find a solution that satisfies both of our inequalities. And that’s because the question says solve 10𝑥 plus one is less than 𝑥 minus eight and 𝑥 minus one is greater than negative five.

So to help us understand the solution we’re gonna give, I’ve drawn a number line. And we’re going to represent each of our inequality solutions on this number line. So first of all, I’ve shown the solution 𝑥 is less than negative one, and this is the solution to 10𝑥 plus one is less than 𝑥 minus eight.

As you can see, I’ve drawn an open circle at negative one and then drawn a line to the left showing that it’s less than this value. The reason it’s an open circle is because it’s 𝑥 is less than negative one, not less than or equal to. If it was less than or equal to, we’d have to color in this circle.

So now I’ve drawn the solution to 𝑥 minus one is greater than negative five, and that solution is 𝑥 is greater than negative four. Again, I’ve got an open circle because it’s just greater than, not greater than or equal to. But this time, the arrow’s going to the right, and that’s because it’s greater than.

Well, if we want to find the solution that satisfies both of our inequalities, then we can see here that, actually, we can get rid of these two sections I’ve colored in blue. So therefore, I’ve shown the solution to our inequalities on the number line in blue. And thus, we can say that if we’re gonna solve 10𝑥 plus one is less than 𝑥 minus eight and 𝑥 minus one is greater than negative five, the solution will be 𝑥 is greater than negative four and less than negative one, as we showed on our number line.

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