Solve each inequality. Give the solution set in interval notation...

Question

Solve each inequality. Give the solution set in interval notation. | 7 - 3x | ≤ 4

Accepted Answer

Hello everyone in this video. We're going to be looking at this practice problem where we want to find the solution set for the absolute value inequality of absolute value of 6 -3 x less than or equal to 54. So in this case we're going to be working with an absolute value. So we want to think about what the absolute value is doing to the components inside the absolute value which in this case is six minus three X. So six minus three X could really be positive. Which in that case it would be positive six X or 6 -3 x Less than or equal to 54 but -3 X could also be negative because recall that the absolute value makes Negative numbers positive. So we could really have negative 6 -3 x less than or equal to 54. So in order to find the solution set, we want to solve for the values of X. That make these inequalities true. So we want to isolate X in each of these inequalities. So if we do that I'll start with if six minus three expose positive and I'll subtract six from both sides. So I have negative three X. Less than or equal to 48 because 54 minus six is 48. And to completely isolate X. Well divide by negative three but I'm dividing a negative across an inequality. So it switches directions from less than or equal to two greater than or equal to So X is greater than or equal to 48, divided by -3 is -12. Sorry, negative 16. So that simplifies the inequality or if the component inside the absolute value were positive and now we want to look at if they were negative. So if they were negative we have this negative on the outside. So in order to simplify this, I'm just going to go ahead and divide or not but I'll distribute it inside this parentheses. So negative one times six is negative six and negative one times negative three X is positive three X. And we still have less than or equal to 54 but now I'm going to be adding six to both sides. So three X less than or equal to 54 plus six is 60. And in order to completely isolate X all divide three. So that means the X is less than or equal to 60 divided by three which is 20. And now we've solved both inequalities and now we want to try to convert these inequalities into an interval so we can get our solution set. And in order to make it easier to see this interval, I'm going to go ahead and Write these on a number line, I'll say that this line here is negative 16 and this line here is 20 And I'll add to zero just to make it clear that we're going from negative numbers to positive numbers. So if I look at my first interval I have X is greater than or equal to negative 16, which means that negative 16 is a lower boundary because X is going greater than that. So I'll draw a bracket and a line going to the right to symbolize that X is greater than -16. And my second interval is saying that X is less than or equal to 20, which means that 20 is an upper boundary. So I'll do a bracket at 20. again a bracket because there's an equal to component, but now X is less than 20, which means I need to go to the left. So I'll draw a line going to the left And it extends all the way to the left. But for my solution set, I need to find where these inequalities overlap. And if we look at the number line, you can see that the lines overlap in between these two boundaries of -16 and 20. So now I can write that in interval notation. So my lower boundary In this case is -16. And from -16. Both of the inequalities overlap until I get to 20. And again both of these have brackets because there's an equal to so bracket negative 16-20 bracket becomes my interval for this solution set. And you can see that answer choice D also has that same interval. So hopefully this video helped and see you next time

Solve each inequality. Give the solution set in interval notation. | 7 - 3x | ≤ 4

Key Concepts

Absolute Value Inequalities

Solving Linear Inequalities

Interval Notation

Watch next