In all exercises, other than exercises with no solution, use inte...

Question

In all exercises, other than exercises with no solution, use interval notation to express solution sets and graph each solution set on a number line.

In Exercises 27–50, solve each linear inequality.

-9x ≥ 36

Accepted Answer

Hello today we're going to be solving the following inequality and we're going to represent it on a number line and give the solution an interval notation. So what we are given is negative seven X is greater than or equal to 56. Now in order to solve for X in this case all we have to do is divide both sides of the inequality by negative seven. But you have to be very careful here. Whenever you multiply or divide an inequality by a negative number, you must change the position of the sign. So when we divide this inequality by negative seven, what we have is X is now less than or equal to negative 56/7, which is going to give us eight. So this is going to be the simplified version of the given inequality. Now we want to go ahead and graph this on the number line. So let's go ahead and first draw the number line. And we're going to put negative eight in the middle of the number line. What this inequality is saying is that we want all the numbers X to be less than negative eight and it can also equal to negative eight. So since we want all the numbers that come before negative eight, we want all the numbers to the left of negative eight on the number line. Or in other words we want all the numbers that approach negative infinity. And since X is allowed to equal to negative eight, we can include negative eight in our solution set. So we can go ahead and represent that as a solid dot on the number line. And since we want all the numbers that come before negative eight, we're gonna shade in the whole left hand side of this number line. And this is going to be the graph of our given inequality. Now let's go ahead and represent the solution in interval notation. We first start by writing the lowest number in our solution set. And in this case here, the lowest number is going to be negative infinity. Since negative infinity doesn't have a finite value to it. We're going to assign a parentheses to it. And we're gonna close this off by writing -8 because it's the last number in our solution set. And since we're including negative eight in our solution set, we're going to close it off with a bracket. And this is going to be the interval notation of the graph of the inequality. And the option that's given to us to accurately represent both of these is going to be a So the solution to this problem is a So I hope this video helps you in understanding how to approach this type of inequality problem. And I'll go ahead and see you all in the next video

In all exercises, other than exercises with no solution, use interval notation to express solution sets and graph each solution set on a number line. In Exercises 27–50, solve each linear inequality. -9x ≥ 36

Key Concepts

Linear Inequalities

Solving Inequalities

Interval Notation

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