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Solve Multi-Step Inequalities - Example 4

In mathematics, a linear inequality is an inequality involving a linear function. Linear inequalities are algebraic equations involving the addition, subtraction, multiplication, and division of numbers and variables, and the four basic arithmetic operations (addition, subtraction, multiplication, and division) performed on them. Linear inequalities can be represented graphically using a closed curve (a line segment joining two points) or open curve (a line segment with no end-points).


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So in this problem, we're being asked to solve the given inequality. And then we're gonna draft the solution set on the number line. Well, I see we have prophecies here, so the first thing we need to do is used to distributive property. So we're going to start by distributing the one house. Well, one half times six x is three X and then we'll have one half times three, which is positive. Three halves. Then we'll bring down our in the quality sign. Greater than or equal to. Now we need to distribute to two thirds. Well, two thirds times nine x is going to be six x and two thirds times negative. One is negative. Two thirds. So now we've removed our parentheses, but we still have fractions here, so we want to get rid of those. Next. Remember to get rid of your fractions. You want to multiply both sides of your inequality by the least common denominator. Well, the least common denominator for three halves and two thirds is going to be six. So we're going to go ahead and multiply both sides of our inequality by six. So let's go ahead and distribute well six times three x is 18 x Then we'll have six times three halves, which is positive night. We'll bring down there in the quality site. Now, for the right hand side, we have six times six X, which is 36 X and then we have six times negative two thirds, which is going to be negative for so great. Now we're gonna rid of our fractions. So the next thing we need to dio is to collect are variable terms on one side of our inequality. So they do this, I'm gonna move to 18 next to the right hand side. So I'm going to subtract 18 x from both sides are right inequality. And I'm doing this because it's the smaller of the two. So that way I'm gonna be left with a positive result. So 18 X's cancel. So we're gonna bring down the nine that's on the left hand side. It's greater than or equal to well 36 x minus 18 X is 18 x and we'll bring down the minus four. Now we have a two step. So to solve for X, we're going to start by adding four to both sides of inequality. We have nine plus four. Well, that's 13. So we have 13 is greater than or equal to 18 X and then to solve for X for now, going to divide both sides of her inequality by 18. Well, 13/18 can't be reduced. So we're gonna bring that down. So we have 13/18 is greater than or equal to X. So perfect. Now we've solved for X. Well, we also need to graph our solution set on a number line. So I'm going to set up our number one. I'm gonna put our key value 13/18 on it, then because X or sorry, 13/18 is a solution because it's greater than or equal to X. That tells us we're gonna have a close circle here. And now it comes to whether we should say the right hand side, the left hand side. Well, remember, we always like to have our variable first. So we were the flip are inequality around? We will be left with X is less than or equal to 13/18. So now that we know X has to be less than this value that tells us we're going to shade the left hand side are our number line. So now we have solved and graft the solution set to our original inequality.