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A rectangular floor can be covered completely with tiles that each measure one square foot. The length of the floor is 1 foot longer than the width and the area is less than 56 square feet. What are the possible dimensions of the floor?

Thus the width is between 0 ft and 7 ft (because the width has to be positive) and the length is between 1 and 8 ft.

Algebra

Chapter 1

THE INTEGERS

Section 8

Quadratic Inequalities

The Integers

Equations and Inequalities

Polynomials

Campbell University

Oregon State University

Baylor University

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So we have a rectangle and we know that the length of the floor is one foot longer than the wits. So wait is one foot longer than the wit. So one place, the wit, we know that the area is less than 56 square feet, so the area is less than 56. Not gonna worry about units right now, So the question for area is area equals length, times with and I want a substitute. This equation here in a thrilling so area equals one plus w. So suddenly you and I want to substitute this thing for area here. Solo one plus w times of you. It's less than 56. So now we have an inequality. The left hand side of learned ish of you. So have W closed. W squared is less than 56. And because we have a w swear I know that we're gonna have a try an army. Also, I'm gonna get all my numbers to one side. Well, there's a chart 50 suit. We're gonna have a lesson zero over here. And that way we can back there looks hand side, you know that we want them to multiply to negative 56 multiplier distant nous. And let's try to get Nick to 56. And we know you want them to add so positively the list of backers of 56 1 in 56. So in 28 4 14 and 78 So because it's negative, we know that only one number in each pair asked me, and I do. So we're gonna eliminate this choice because we needed to add to one. We're gonna eliminate this also, And this here so he knows that maybe 78 and the seventh can be negative. So this is our pair here. We're gonna use that too. But apart our middle turn, so have W squared minus seven w plus eight w and bring down to minus 56. So a group the first to grip the 2nd 2 and then factor sir, from diverse. When we get back to out of W and we're left with W minus seven from the cycle would get back around eight. So we're left with W minus seven. So because this factor in this factor of the same, we know they will have w plus eight time still view minus seven is less than zero. So we have our two back tears and are in the quality. Solving is different thing solving for a regular equation, as if they were said equal to zero. It's different because inequality gives us a range of values instead of business. That number. So what he says list, then that just means negative less than zero means negative and the only way to get a negative product. We can have this factory positive this factor b negative, and I will give us a negative product. But we also need to find if this factor one negative in this factory positive and I will also give us a negative product. So we need to solve for both of these situations here, something the 1st 1 will have W Plus eight is positive or greater than zero, and W minus seven is less than zero or negative, and then we just combine. The result does the track eight from both sides here will have w is greater than negative eighths. Then we add setting the boat sides Here you'll have ws Listen seven. Look, John I'm relying will have negative eight impulsive seventh the W is less than seven. So to go this way and w is greater than negative bait. So you go this way. So, to shade the common area No, you is in between negative Bates and seven. So, yes, I would combine it here. But we also need to saw four second situation for a second situation is if the 1st 1 is negative and the second is positive and combined results there, So w plus eight will be negative. Listen, zero and W minus seven will be positive or greater than zero. That's all for W. We get W. It's less than negative. Eight. Dance off the w Here we get w is good and positive. Seven. No joking and realize will have negative eights and positive. Seven. The W is greater than seven. So to go that way and w is less than that innovates sort of it that way. And there is no common area between the suit, so there's no way to combine our second situation here. Therefore, our answer. ISS negative eight. His list and W He's lost 37 so it wants to know what a possible dimensions of the floor for a possible dimensions are obviously going to be in this range here that we just brute. However, we cannot have a negative wit. So our wits he's going to be between zero in seven because we cannot have a negative. I'm gonna write that down. Cannot have negative would. So we're gonna properly 0 to 7 feet. We also need to find a link, and we're just gonna plug w in for Are the buttes blindly? So originally, our equation ways bill equals one plus w. And if we just plug W in our lowest w so we can plug in zero and we get one for our lowest value. We also need to find the highest value. So we have one plus in our highest valuable study. So it's a so link is in between wine and a So here. That's the answer is here. And you saw further possible dimensions. I'll always remember whenever dealing with, like, a door be or anything of the sort. Remember that you cannot have a native value

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