Question

3 Read the statements given below and set up an equation.
(i) Half of a number added to 10 is 15.
(ii) One-fifth of a number minus 4 gives us 3.
(iii) One-third of a number added to itself gives 10.
(iv) Two more than three times the difference between eight and number
x gives 0.
(v) One-third of a number plus 2 is 4.
(vi) 7x subtracted from 16 gives 2.
(vii) If we take 3 from 5 times p, then we get 7.
(viii) 6 added to 3 times m gives 21.




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Answer 1

Answer:

Problems on previous material

1. Answer the following problems in the textbook: A.7.1, A.7.3, A.8.2,

A.8.5, 1.9.2, 1.9.6, 1.10.3, 1.10.5

Solution to A.7.1. Let n = 3. For this n, the equation 4x

2+x−n =

0 becomes 4x

2 + x − 3 = 0. This equation has a rational root x = −1.

Solution to A.7.3. The converse: “Every continuous function is

differentiable” (this is false — e.g., the function f(x) = |x| is continuous but not differentiable). The contrapositive: “Every function that

is not continuous is also not differentiable”. This is true.

Solution to A.8.2. I claim that 1 + 3 + 5 + . . . + (2n − 1) = n

2

.

The proof is by induction on n. For n = 1, the left-hand side is 1, and

n

2 = 1, so the statement is true. Assume that it is true for n−1. This

means that

1 + 3 + 5 + . . . +

2(n − 1) − 1

= 1 + 3 + 5 + . . . + (2n − 3) = (n − 1)2

.

It follows, using the induction hypothesis, that

1 + 3 + 5+ . . . + (2n − 1) = (1 + 3 + 5 + . . . + (2n − 3)) + (2n − 1)

= (n − 1)2 + (2n − 1) = n

2 − 2n + 1 + (2n − 1) = n

2

Step-by-step explanation: