Solve each inequality. Give the solution set using interval no...

Question

Solve each inequality. Give the solution set using interval notation. See Examples 8 and 9. 4x + 7 ———— ≤ 2x + 5 -3

Accepted Answer

Welcome back everyone. In this problem, we want to find a solution set for the inequality nine X plus 22 all divided by negative five is less than or equal to six X plus 19. And we want to express our answer in interval notation for our answer choices A is open parenthesis negative three infinity, closed parentheses. B is open parenthesis, negative infinity, negative three closed bracket. C is open parenthesis, negative infinity three closed bracket and D is open bracket negative three infinity closed parentheses. So if we're going to figure out the solution set, then we have to solve our inequality and recall that we can solve an inequality with the same logic that we use to solve an equation. So if I'm going to get X by itself first, I need to get rid of the negative five and I can do that by multiplying both sides by negative five. Now if when I multiply both sides of an inequality by a negative number, then recall that we have to flip the inequality sign so less than our equal to no becomes greater than or equal to negative five, cancels negative five. So that leaves me with nine X plus 22. On the left hand side, while on the right hand side, I have to distribute negative five. So negative five multiplied by six X is negative 30 X and negative five multiplied by 19 is negative 95. Now to get X by itself, I can subtract 22 from the left hand side. And when I do that 22 minus 22 that leaves us with nine X and negative 30 X minus 95 minus 22 is minus 117. I still have X on both sides. So to get X on the left hand side, only I can add 30 X to both the right hand side and the left hand side of the inequality. And no nine X plus 30 X is 39 X while negative 30 X plus 30 X equals zero. Therefore 39 XE is greater than or equal to negative 117 to get X by itself. Finally, I can divide by three and by rather and 39 divided by 39 users with X. While on the left, on the right hand side of our inequality, negative 117 divided by 39 equals negative three. Therefore X is greater than or equal to negative three. So we have our solution set. Now we just need to represent it in interval notation. So let's do that. So if X is greater than our equal to three. Let me use our number line to help us. And we are saying that at negative three X would be greater than, or equal to that. So it would be more than negative three and it could also be negative three. So if it includes our value, if it's greater than our equal to, we use our bracket, so it would be open bracket, negative three and it's going on to all everything greater than negative three. So it's continuing to infinity. So it would be infinity, close parenthesis because it never will become infinity. Therefore, d sorry, our solution is answer choice. D thanks a lot for watching everyone. I hope this video helped.

Solve each inequality. Give the solution set using interval notation. See Examples 8 and 9. 4x + 7 ———— ≤ 2x + 5 -3

Key Concepts

Solving Rational Inequalities

Critical Points and Sign Analysis

Interval Notation and Domain Restrictions

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