Solve the following inequality: negative nine 𝑥 plus 14 is greater than or equal to negative 20 minus eight 𝑥.
When you multiply or divide each side of an inequality by a negative number, the direction of the inequality symbol must be reversed for the inequality to remain true. So it’ll be good to keep this in mind since there are so many negative signs in our inequality.
So the first thing we should do is to get the 𝑥s on one side and the constants or the normal numbers on the other side. Now with an inequality the variables are usually on the left-hand side because you read it left to right. So let’s go ahead and place our variables on the left-hand side of the inequality and our constants on the right-hand side of the inequality sign, which is this greater than or equal to sign.
So we would add eight 𝑥 to both sides of our inequality. On the right-hand side, they would cancel and on the left, negative nine 𝑥 plus eight 𝑥 is negative one 𝑥. And now we bring down the 14, our inequality sign, and the negative 20. So now we need to bring the 14 over to the right-hand side. So we need to subtract 14 from both sides; that’s how we will get rid of positive 14.
So on the left-hand side, they cancel and on the right, negative 20 minus 14 is negative 34. Now notice we need to get rid of this negative one that’s been multiplied to 𝑥. So we have to divide both sides by a negative number. And as we said before, when you multiply or divide each side of an inequality by a negative number, the direction of the inequality symbol must be reversed; this way the inequality we would still remain true.
So since we are dividing by a negative one, our greater than or equal to sign will change to less than or equal to. And negative 34 divided by negative one will be positive 34 because two negatives make a positive. Therefore, 𝑥 is less than or equal to 34.