SOLUTION: 1.) Sketch the graph of{(x,y)absolute value x-y=3 and x+2y>4}
2.) Sketch the graph of absolute value of x + absolute value of y=1
3.)A and B working together can do a job in 6 an
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Question 208203This question is from textbook ALGERA
: 1.) Sketch the graph of{(x,y)absolute value x-y=3 and x+2y>4}
2.) Sketch the graph of absolute value of x + absolute value of y=1
3.)A and B working together can do a job in 6 and 2/3 hours. A become ill after 3 hours of working with B, and B finished the job, continuing to work alone in 8 and 1/4 hours. How long would it take each working alone to do the job?
This question is from textbook ALGERA
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
1.) Sketch the graph of{(x,y)absolute value x-y=3 and x+2y>4}
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before you can graph them, you have to solve them.
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|x-y| = 3
this means that:
x-y = 3
or
x-y = -3
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solving for y, we get:
y = x-3
or
y = x+3
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to show that this is true, take any value of x and substitute in the equation.
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for the first part of the equation, we get:
if x = 5, then y = 2 and |x-y| = |5-2| = |3| = 3 which equals 3.
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for the second part of the equation, we get:
if x = 5, then y = 8 and |x-y| = |5-8| = |-3| = 3 which equals 3.
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the graph of this equation would be:
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the top line is x+3 and the bottom line is x-3.
keep in mind that the value of y is x+3 OR x-3. it can only be one or the other. it can never be both at the same time.
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your second equation is:
|x+2y| > 4
this means that:
x + 2y > 4
or
x + 2y < -4
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solving for y we get:
y > (4-x)/2
or
y < (-4-x)/2
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to confirm that this is correct, take any value of x and see if it works.
let's take x = -20
for the first part of the equation, we get:
y > (4-x)/2 which becomes:
y > (4-(-20))/2 which becomes:
y > 24/2 = 12
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for the second part of the equation, we get:
y < (-4-x)/2 which becomes:
y < (-4-(-20))/2
y < 16/2 = 8
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so if x = -20 and (y is > 12 or y is < 8), the equation should be satisfied.
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let's take x = -20 and y = 13
since 13 > 12, this should satisfy the first part of the equation.
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|x+2y| > 4 becomes |(-20) + 26| = |6| = 6 > 4 which is true.
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let's take x = -20 and y = 7
since 7 < 8, this should satisfy the second part of the equation.
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|x+2y| > 4 becomes |(-20) + 14| = |-6| = 6 > 4 which is true.
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if we took y = 10, we would have gotten:
|x+2y| > 4 becomes |(-20) + 20| = |0| = 0 > 4 which is NOT true.
if we examine y we see that is is not > 12 and it is not < 8 which violates the rules and yields the false answer.
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when x = -20, if y > 12 or y < 8, then the equation will be satisfied.
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graph of these equations would be:
the slanted lines are the lines of your equations.
the area of acceptable range is outside of the area between the 2 slanted lines.
when x = -20, y > 12 or < 8.
when x = 0, y > 2 or < -2
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2.) Sketch the graph of absolute value of x + absolute value of y=1
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again, you have to solve the equation before you can sketch the graph.
solving for y, we get:
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y = +/- (1-|x|)
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that little area between [-1 <= x <= 1], and [-1 <= y <= 1] is the acceptable domain for x and range for y. anything outside of that is invalid.
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3.)A and B working together can do a job in 6 and 2/3 hours. A become ill after 3 hours of working with B, and B finished the job, continuing to work alone in 8 and 1/4 hours. How long would it take each working alone to do the job?
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A and B working together can do a job in 6 and 2/3 hours which is the same as 20/3 hours.
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A become ill after 3 hours of working with B, and B finished the job, continuing to work alone in 8 and 1/4 hours.
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since they worked together for 3 hours, this mean that they completed 3 hours out of what would have been 20/3 hour job.
3 hours divided by 20/3 hours means they finished 9/20 of the job which means that 11/20 of the job remained.
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B completes 11/20 of the job in 8 and 1/4 hours which is the same as 33/4 hours.
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We know that rate per unit * time = number of units produced.
the units produced here is 1 job.
we let a = number of units per hour that A can produce.
we let b = number of units per hour that B can produce.
our formula is:
(a + b) * 20/3 = 1
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the formula for B finishing the job by himself is:
b * 33/4 = 11/20 * 1 = 11/20
solving for b, we get: b = (11/20) / (33/4) = (11 * 4) / (20 * 33) = 1 / 15
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this means that B can complete (1/15) of the job in 1 hours which means that B would take 15 hours to complete the job.
formula for B doing the job alone would be:
(1/15)*T = 1
solving for T gets T = 15 hours which is the time it would take B to do the job alone.
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now that we have B's production rate, we can solve for A's production rate.
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we go back to the original equation that states:
(a + b) * 20/3 = 1
we multiply out the factor to get:
(20/3)*a + (20/3)*b = 1
we replace b with (1/15) to get:
(20/3)*a + (20/3)*(1/15) = 1
we multiply both sides of this equation by (3)*(15) = 45 to remove the denominators and we get:
300*a + 20 = 45
we subtract 20 from both sides to get:
300*a = 25
we divide both sides by 300 to get:
a = 1/12
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the number of units that A can produce per hour is 1/12
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if A can produce 1/12 units per hour, then A can produce 1 unit in 12 hours.
the formula for A would be:
(1/12) * T = 1
solve for T gets T = 12 hours
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it would take B 15 hours to complete the job working alone.
it would take A 12 hours to complete the job working alone.
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to confirm the answer as correct, we solve the original equation of:
(a + b) * 20/3 = 1
this becomes:
(1/15 + 1/12) * 20/3 = 1
multiply both sides of this equation by (12)*(15) to get:
(12 + 15) * 20/3 = (12)*(15)
this becomes:
(27)*(20)/3 = (12)*(15)
this becomes:
9*20 = 12*15
this becomes:
180 = 180 confirming the values for a and b are correct.
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second confirmation is the equation for B to complete 11/20 of the job.
since he finished the job in 33/4 hours, and his rate of production is 1/15 units per hour, the equation for him would be:
1/15 * 33/4 = 11/20
multiply both sides of this equation by 15*4*20 to remove the denominators and you get:
20*33 = 4*15*11
this becomes:
660 = 660 confirmning the rate of work for B is accurate.
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the key to solving this was to understand how much of the job was left to do and to use that fact to solve for the unit production rate of B.
once you got that, you could then solve for the unit production rate of A.
once you got those, you could then solve for how long it would take each person to complete the job alone.
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