a Let x the number of days and y be the number of pounds. Notice that since one gallon of water weighs 8 pounds, 0.5 gallon of water weighs 4 pounds.
B
b *Marc will run out of supplies at the point where y=9x+20 intersects the line y=50.
Jessica will run out of supplies at the point where y=7x+10 intersects the line y=35.
C
c Interpret the result that you found in Part B.
A
aInequalities:
y≥ 7x+10
y≥ 9x+20
y≤ 35
y ≤ 50 Graph:
B
b 3.3 days
C
c Marc will run out first.
Practice makes perfect
a Let x the number of days and y be the number of pounds. We will first write an inequality that represents the number of pounds depending on the number of days for both of them. Notice that since one gallon of water weighs 8 pounds, 0.5 gallon of water weighs 4 pounds. Let's start!
Jessica
Marc
Verbal Expression
Algebraic Expression
Verbal Expression
Algebraic Expression
Weight of the equipment (lb)
10
Weight of the equipment (lb)
20
Total weight of the food and water for x days (lb)
7 x
Total weight of food and water for x days (lb)
9 x
Total weight of the supplies is less than or equal to y pounds
y≥ 7 x+ 10
Total weight of the supplies is less than or equal to y pounds
y≥ 9 x+ 20
We also know that Jessica is capable of carrying 35 pounds and Marc can carry 50 pounds. This means that Jessica's load will be less than or equal to 35 pounds and Marc's load will be less than or equal to 50 pounds. Thus, we have four inequalities to write a system.
y≥ 7x+10
y≥ 9x+20
y≤ 35
y ≤ 50
Let's begin by graphing the first inequality. To do that will first determine its boundary line.
&Inequality I &&Boundary Line I
&y ≥ 7x+10 &&y = 7x+10
Notice that the boundary line is in slope-intercept form, so we can immediately determine its slope and y-intercept.
Slope-Intercept Form
y= 7x+ 10
We can plot the y-intercept and then find a second point by using the slope to draw the line.
Now we can draw the line. Be careful that the number of days and pounds cannot be negative, so the line will be bound by the axes. It will also be solid because the inequality is non-strict.
To complete the inequality, we will test a point so that we can decide which region we should shade. Let's test (0,0).
The point did not satisfy the inequality. Thus, we will shade the region that does not contain the point.
Proceeding in the same way, the second inequality can also be graphed.
&Inequality II &&Boundary Line II
&y ≥ 9x+20 &&y = 9x+20
We can again test the point (0,0).
By graphing the last two inequality, the graph of the system can be completed.
ccc
& &Inequality II &&Boundary Line II
&III: &y ≤ 35 &&y = 35
&IV: &y ≤ 50 &&y = 50
Both boundary lines are horizontal lines that pass through the point (0,35) and (0,50). Both inequalities indicates the points below the lines as solutions. Therefore, we will shade below the boundary lines.
The overlapping section is the solution set to the system.
b Let's assume Jessica only uses the supplies she has carried, and Marc only uses the supplies he has carried. Then Marc will run out of supplies at the point where y=9x+20 intersects the line y=50. By solving the these two equations together, we can find the point.
This point is (3.3,50) so Marc will run out of supplies after 3.3 days. Jessica will run out of supplies at the point where y=7x+10 intersects the line y=35. Let's find this point.
