Match each equation or inequality in Column I with the graph o...

Question

Match each equation or inequality in Column I with the graph of its solution set in Column II. | x | ≤ 7

List of absolute value equations and inequalities matched with number line graphs showing solution sets between -7 and 7.

Accepted Answer

Everyone in this problem, we're asked to graph the solution set for the following inequality. And we have that the absolute value of X is less than or equal to six. Yeah. All right. So when we have an absolute value in equality, OK, we have the absolute value of X is less than, or equal to six. And we can think about breaking this absolute value into two different situations. We know that if X is positive, OK. So if X is greater than or equal to zero, then the absolute value of X is just equal to X and the absolute value is gonna give us a positive value. So if X is already positive, we're just getting that same X value F. So if the absolute value of X is just equal to X, then we get the inequality that X is the less than or equal to six. Now we have to kind of combine these two values, we have that X has to be greater than, or equal to zero for this to be true. And if it is, then we get X is less than or equal to six. And so for this problem or this situation, we have the zero is less than, or equal to X is less than, or equal to six. So X is positive, it's bigger than or equal to zero, but it has to be less than six to satisfy the inequality. OK. Now moving to the second situation and the second case, if X is less than zero and then what's inside the absolute value will be negative in order to make it positive for that absolute value function we multiply by a negative. OK? And so the absolute value of X is gonna be equal to negative X, right X is negative, we multiply by a negative. Now we have a positive value which is what we want. And so our inequality becomes that negative X is less than or equal to six if we divide by negative one. OK? Remember that the inequality has to flip because we're dividing by negative and we get that X is greater than or equal to negative six. OK? So putting our two values together X is less than zero and it's greater than or equal to negative six. We forget that negative six is less than or equal to X which is less than zero. And so we have these two cases and now we can put them together. On the left hand side, we have everything from zero up to and including six. On the right hand side, we have everything from negative six up to zero. While we can combine those and write this as the negative six is less than or equal to X, which is less than or equal to six. And you may see your textbook or your professor, they might have given you this already. OK? Where you can go directly from the absolute value of X less than or equal to six into this inequality, negative six is less than or equal to X is less than or equal to six. OK. That might be something that you can do directly. And I just broke up those two cases to show you why. That's true. All right. So now we want to draw this on our diagram. We have our number line here and we wanna include all values between negative six and six including the end points. So at the end points, we're gonna draw close c circles to indicate that we're including those points. So that's gonna be at negative six and six. And between those values, we're just gonna connect them with a line to indicate that all points in between are solutions as well. OK. So that's the graph where our solution set for this problem. Thanks everyone for watching. I hope this video helped see you in the next one.

Match each equation or inequality in Column I with the graph of its solution set in Column II. | x | ≤ 7

Key Concepts

Absolute Value Inequality

Interval Notation and Graphing

Matching Equations to Graphs

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