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# 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 they were graft. The solution set on the number one. Well, I see we don't have any prophecies, so we don't have to worry about that. But we do have fractions. So remember to remove fractions, we're going to multiply both sides of our inequality by our least common denominator. Well, the least common denominator for two thirds and 1/4 is going to be 12. So we need to multiply both sides of our inequality by positive 12. So now let's go ahead and distribute Well, 12 times two thirds X is going to be eight X. Then we're gonna have 12 times negative one, which is negative. 12. We'll bring down our in the quality sign greater than or equal to. Then we'll distribute the 12 on the right hand side. Well, 12 times one for effects is really just three X and then we'll have 12 times three, which is positive. 36. Perfect. Now we've gotten rid of our fractions, So the next thing we need to dio is collect are variable terms on one side of our equation. So to do this I'm going to subtract three x from both sides of our inequality. Well, eight X minus three X is five x so we're left with five. X minus 12 is greater than or equal to 36. So now we just have a two step. So we're going to start by adding 12 to both sides of our inequality. So we're gonna have 36 plus 12, which is 48. So we're left with five X is greater than or equal to 48. And then we'll divide both sides of our inequality by five. And because we're divided by a positive number, we don't need to change our sign. So we have X is greater than or equal to. Well, we can't reduce for the over five so we can leave that as our answer for the 8/5. Let me write that five. So it's a little bit more legible for us. Okay, so now we've solved our inequality. So we found that our answer is that X would have to be greater than or equal to 48 5th. So now what we want to dio is graft, a solution set on a number line so to do that, we're going to start by making our number line. We're gonna put our key value, which in this case is 48 5th underneath. And in this case, we're going to use a close circle because our inequality sign is greater than or equal to, Which means that for the 8/5 is a solution, and because our inequality says X is greater than equal to 48 5th, all the values that would be greater than it will occur to the right hand side of this point. So we're going to shade the right hand side of our inequality. So now we have solved and graph the solution set for the given inequality.

Syracuse University