a Based on the given table, we need to determine if the solution to the inequality will be of the form yc, where c is a real number.
y
7y+2
<
6(4-y)
0.5
7(0.5) +2 =5.5
True
6(4-0.5)=21
1
7(1) +2 =9
True
6(4-1)=18
1.5
7(1.5) +2 =12.5
True
6(4-1.5)=15
2
7(2) +2 =16
False
6(4-2)=12
Notice that the inequality is true for the first 3 values and then it becomes false. Because the inequality is true for the values 1.5 and below, values less than 1.5, we know that the inequality will be of the form y
b In Part A, we determined that the solution to the inequality will be of the form y
y
7y+2
<
6(4-y)
0.5
7(0.5) +2 =5.5
True
6(4-0.5)=21
1
7(1) +2 =9
True
6(4-1)=18
1.5
7(1.5) +2 =12.5
True
6(4-1.5)=15
2
7(2) +2 =16
False
6(4-2)=12
The greatest value that satisfied the inequality was 1.5, since 2 was false. This means that the value of c should be between these two numbers. One way to estimate the value is by taking the average.
Therefore, an estimation that we could use for c is 1.75.
c In Part B, we found the estimated solution to be y<1.75. Now, let's solve the inequality and compare the actual solution with our approximation. We can solve the inequality by isolating y using inverse operations and the Properties of Inequality.
The exact solution is 2213. We can express the solution in decimal form.
y < 2213 ⇒ y<1.6923 ...
If we take the difference between the exact solution and our estimated value, we can compare them to see how close we were.
