Video: Solving Systems of Linear Inequalities in One Variable

Solve −5 − 3𝑥 ≥ 4 and 𝑥 + 3 < −5.

03:50

Video Transcript

Solve negative five minus three 𝑥 is greater than or equal to four and 𝑥 plus three is less than negative five.

So in order to actually solve this problem, what we’re gonna do is actually solve each of our inequalities separately first. So we’re gonna start with negative five minus three 𝑥 is greater than or equal to four. And when we solve an inequality, we do it in much the same way as when we’re solving an equation.

So the first step we’re gonna do is actually add three 𝑥 to both sides of the inequality. And that’s because we would like a positive 𝑥 cause it’s easier to deal with. So therefore, we get negative five is greater than or equal to three 𝑥 plus four. Then next, we’re actually gonna subtract four from each side of the inequality, because we want the 𝑥 on its own.

So when we do that, we’re gonna get negative nine — and that’s because negative five minus four is negative nine — is greater than or equal to three 𝑥. And then, finally, what we’re gonna do is actually divide each side of our inequality by three. And when we do that, we’re gonna get negative three on the left-hand side and just 𝑥 on the right-hand side. So we can say that 𝑥 is less than or equal to negative three.

So now what we’re gonna do is actually solve our other inequality. And that inequality is that 𝑥 plus three is less than negative five. So this time, to leave 𝑥 on its own, what we’re gonna do is actually subtract three from each side of the inequality. And when we do that, we get 𝑥 is less than negative eight. That’s because if you subtract three away from negative five, it goes to negative eight, cause we actually take three steps left on our number line, so we go from negative five, negative six, negative seven, to negative eight.

So now we’ve actually got a solution for both of our inequalities. What we need to do is actually see how we’re gonna find a solution that will actually satisfy both of our inequalities, because the question says solve negative five minus three 𝑥 is greater than or equal to four and 𝑥 plus three is less than negative five.

Well, to help us actually understand what the answer is going to be, I’ve drawn a number line. And we can actually show both of our inequalities on this. So first of all, I’ve actually shown the inequality that satisfies negative five minus three 𝑥 is greater than or equal to four, because what I’ve drawn here is 𝑥 is less than or equal to negative three. That’s because I’ve drawn a dot on negative three and then drawn an arrow to the left to show that it’s less than.

But also the dot is colored in. And the reason it’s colored in is because of this line here. And this line on our inequality notation shows us that it’s actually less than or equal to, and it’s that “or equal to” which means our dot is colored in.

Okay, so now let’s represent the other inequality on the same number line. Well, I’ve now drawn on the other inequality, and I‘ve actually drawn an open circle at negative eight, because it’s 𝑥 is less than negative eight. And we’ve done an open circle because this one is just less than; it’s not less than or equal to.

Well, we can see that if want to satisfy both of these inequalities, we can disregard the portion that’s actually greater than negative eight, as the only part that can satisfy both of our inequalities is the part that is less than negative eight. So therefore, we can say that the solution that satisfies negative five minus three 𝑥 is greater than or equal to four and 𝑥 plus three is less than negative five is 𝑥 is less than negative eight. And we’ve shown it here on the number line as well.

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