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# Writing two-variable inequalities word problem

Sal solves a word problem about scores in a chess tournament by creating a two-variable linear inequality.

## Want to join the conversation?

• I knew that there would eventually be a chess question, but I didn't think it would mention Fabi. Magnus Carlsen, Levon Aronian, and Hikaru Nakamura better be used too. : )
• underrated comment
(1 vote)
• What if the question also stated, that the maximum amount of games Fabiano can play is 10?
Would we then need to make a system of equations:
W + 1/2D ≥ 6.5
W + D = 10
And if that's the case, how would we solve this system?
(1 vote)
• If you say max of 10 games, then it would be W+D≤10. The issue with this is that you would actually have to consider 3 variables, (wins, draws, losses), so graphing in 3 dimensions is probably out of the question, thus you have to consider all possible answers. With 7 or more wins, it is obviously true. This would lead to multiple choices (10,0,0)(9,0,0)(9,1,0)(9,0,1)(8,0,0)(8,1,0)(8,1,1)(8,2,0)(8,0,1)(8,0,2), (7,0,0)(7,1,0)(7,1,1)(7,1,2)(7,2,0)(7,2,1)(7,3,0)(7,0,1)(7,0,2)(7,0,3). With 6 wins, it would require at least one draw giving (6,1,0)(6,1,1)(6,1,2)(6,1,3)(6,2,0)(6,2,1)(6,2,2)(6,3,0)(6,3,1) and (6,4,0). With 5 wins, you need at least 3 draws giving points (5,3,0)(5,3,1)(5,3,2)(5,4,0)(5,4,1) and (5,5,0). 4 wins require at least 5 draws giving (4,5,0)(4,5,1) and (4,6,0). 3 wins and 7 draws give (3,7,0) You have to win at least 3 total games.
These are related to triangular numbers with a pattern of 1, 3, 6, 10. Total number of choices would be starting from 10 wins and going down to 3, 1+3+6+10+10+6+3+1=40 possible solutions.
• how do we know which variable is y and which is x (if we're using slope intercept)?
(1 vote)
• In the slope-intercept form of a linear equation, y = mx + b, y is the dependent variable and x is the independent variable.

The variable on the left side of the equation is typically the dependent variable (in this case, y
), and the variable on the right side of the equation is typically the independent variable (in this case,
x).

The coefficient of x (m) is the slope of the line, and b is the y-intercept, which is the value of y when x = 0.

So, in the equation 2x + 8y = 9, y is the dependent variable and x is the independent variable.

Maybe this is what you're looking for..? Hope this helps!!
• is that a fabiano caruana reference??
(1 vote)
• Why isn't the inequality (W + D ≥ 6.5)? The sentence states that .5 = "points for each game that ends in a draw", which = D. Isn't saying .5D being redundant?
(1 vote)
• D doesn't equal 0.5 points automatically so you would need a coefficient. D is just a variable so we can't assume that it has value.
(1 vote)
• I don't understand why we swap a sign when dividing both side by negative value? I know this is a rule but how I can derive it?
• By trial and error: If -x≥5, 0 does not work (0≥5), 5 does not work (-5≥5), but -5 does work (-(-5)≥5), -4 does not work (-(-4))≥5 or 4≥4), -6 and more will all work, so x≤-5.
Second way, if -x≥5, I can add x to both sides to get 0≥x+5, subtract 5 to get -5≥x. Flipping this around gives -x≤-5.
• How do I help my daughter with this: Suppose s and t are two numbers and that s>t. Determine whether each inequality must be true.
A. s + 15 > t + 15
• If s is always Greater Than (>) t, than we can use that info to plug in for those two, so let's use s=6 and t=5, so 6 > 5. (6) + 15 > (5) + 15, so it is true.
• how to do this
• So, if he gets 1/2 point for a draw, why is it written 0.5d? Would one draw give him the half point he needs?

Why isn't the inequality written W+D≥6.5?
• If you use your version of the inequality, it means that every draw is worth 1 point, not 1/2. So, you would get the wrong results.
• Is there any solution for -b <( x-y)b<b?
Or
How to solve it?
• Please clarify - You have multiple variables. What variable were you asked to solve for?

Note: With 3 variables and 1 inequality, you can't get to a numeric answer. You don't have enough info.
(1 vote)

## Video transcript

- [Voiceover] Fabiano wants to score at least 6.5 points in a major chess tournament. He scores one point for each game that he wins, and he scores 0.5 points for each game that ends in a draw. Write an inequality that represents the number of games Fabiano should win and draw, D, to achieve his goal. So I encourage you to pause the video and see if you can do that. Write an inequality in terms of the number of games won, so capital "W", and the number of draws, capital "D", that represents what he needs to do to actually achieve his goal. All right, let's work through it together. So how many points he is going to get from winning? So if he wins "W", he's gonna win W games, and he gets one point for each of them, so it's gonna be one point per game, times the number of games. So one W I could just write as W. So this is the points from winning. From wins, I could say. And what are his points gonna be from the draws? Well, from the draws, he's gonna have D draws, and he gets 0.5 points for each of them. zero point five times D, this is going to be the points from the draws. Now, this right over here is going to be his total points, points from wins, points from draws. I'm assuming he gets no points for losses. And we want this number, the total number of points, his score, to be at least 6.5. So we want this to be greater than or equal to 6.5. If it says Fabiano wants to score more than 6.5, then it would have been greater than. But it says Fabiano wants to score at least 6.5, so that's greater than or equal to 6.5. He's ok if he scores 6.5. And there you have it, we have our inequality in terms of the number of games he needs to win and draw and this inequality needs to be true in order for him to score at least six and a half points in this major chess tournament.