The <em><u>correct answers</u></em> are:
#1) No, because the bike order does not meet the restrictions of 4c + 6a ≤ 120 and 4c + 4a ≤ 100;
#2) E, F and G;
#3) y ≤ −2x + 3
and y ≤ x + 3;
#4) Graph of two lines that intersect at one point. Both lines are solid. One line passes through points negative 2, 2 and 0, 3 and is shaded below the line. The other line passes through points 0, 1 and 1, negative 2 and is shaded above the line.;
#5) (1, -5); and
#6) y ≤ 2x − 2
and y > −x + 3.
Explanation:
#1) The first inequality, representing the number of hours it takes to build the bikes, is 4c+6a ≤ 120. This is because each child bike takes 4 hours to build and each adult bike takes 6 hours to build, and they have up to 120 hours to build bikes.
The second inequality, representing the number of hours it takes to test, is 4c+4a ≤ 100. This is because each child bike takes 4 hours to test and each adult bike takes 4 hours to test, and they have up to 100 hours to test bikes.
Using 20 for c and 6 for a, the first inequality gives us 4(20)+6(6)≤120; 80+36≤120; 116≤120. This is true.
The second inequality gives us 4(20)+4(6)≤100; 80+24≤100; 104≤100. This is not true.
#2) Substituting the x-values in for x and the y-values in for y, we have:
E: 2x + y < 10
and 2x − 4y > 8; 2(4)+-2<10 and 2(4)-4(-2)>8; 8+-2<10 and 8--8>8; 6<10 and 16>8. Both of these are true statements.
F: 2x + y < 10
and 2x − 4y > 8; 2(1)+-6<10 and 2(1)-4(-6)>8; 2+-6<10 and 2--24>8; -4<10 and 26>8. Both of these are true.
G: 2x + y < 10
and 2x − 4y > 8; 2(-3)+-10<10 and 2(-3)-4(-10)>8; -6+-10<10 and -6--40>8; -16<10 and 34>8. Both of these are true.
None of the other points work when substituted in.
#4) The slope of the line through (-2,2) and (0,3) is m=(3-2)/(0--2)=1/2. The y-intercept is one of the points given, (0, 3). This makes the equation y=1/2x+3. Since it is graphed below the line, the inequality is y≤1/2x+3.
The slope of the line through (0, 1) and (1, -2) is m=(-2-1)/(1-0)=-3/1=-3. The y-intercept is one of the points given, (0, 1). This makes the equation y=-3x+1. Since it is graphed above the line, the inequality is y≥-3x+1.
#5) Substituting our x-values in for x and y-values in for y, we have:
y < −3x + 3
and y < x + 2; -5<-3(1)+3 and -5<1+2; -5<-3+3 and -5<3; -5<0 and -5<3. Both of these are true.
#6) The slope of the line through (0, -2) and (1, 0) is m=(0--2)/(1-0)=2/1=2. The y-intercept is one of the points given to us, (0, -2). This makes the equation y=2x-2. Since the graph is shaded below and has a solid line, this makes it y≤2x-2.
The slope of the line through (0, 3) and (3, 0) is m=(0-3)/(3-0)=-3/3=-1. The y-intercept is one of the points given to us, (0, 3). This makes the equation y=-1x+3. Since the graph is shaded above the line and has a dashed line, this makes it y>-1x+3.
