We have, (- 83) x (- 28) = 83 x 28 = 2324

We have, LHS = 39 + (-24) -15 = 39 - 39 = 0
and RHS = 36 + (-52) - (-36)
= 36 - 52 + 36 = 72 - 52 = 20
Since, 0 < 20
39 + (-24) -15 < 36 + (-52) - (-36).

We have, LHS = -3 + 7 -19 = -3 + 7 -19 = - (19 + 3) + 7 = - 22 + 7 = - 15
and  RHS =15 -8 + (-9) = 15 + (-8 -9) = 15 + (-17) = - 2
Since, -15 < - 2
\(\therefore\)   -3 + 7 -19 < 15 - 8 + (- 9)

We have, LHS = (-8) + (-4) = - 8 - 4 = -12
and RHS =(-8) - (-4) = -8 + 4= - 4
Since, -12 < -4
\(\therefore\)  -8 + (-4) < (-8) - (-4)

Correct, when a number and its opposite add up to give zero, they are called additive inverse of each other. In other words, the additive inverse of a number is the opposite of that number.
e.g.

Incorrect, when a positive integer and a negative integer are added, we determine the difference of their absolute values and assign the sign of addend having greater absolute value.
e.g. Let the positive integer be 26 and negative integer be -33.
\(\therefore\) Sum=26+(-33)
= 26 - 33 = - 7 (negative integer)
Again, let the positive integer be 325 and negative integer be - 220.
\(\therefore\) Sum = 325 + (- 220) = 325 - 220 = 105 (positive integer)
Hence, the statement is not correct.

Incorrect, when two negative integers are added, we get a negative integer.
e.g. Let two negative integers be - 46 and - 83.
\(\therefore\) Sum =-46+(-83)=-46-83=-129 (negative integer)
Hence, the statement is not correct.

We have, 7, - 5, 4, 0 and - 4
Since every positive integer is greater than 0 and every negative integer is less than O. So, rearrange the given numbers in ascending order i.e. -5, - 4, 0, 4, 7.
In order to represent the given integers on the number line, we draw a line and mark a point on it almost in the middle (say0) of the line.
Now, starting from 0 and count 4 units on the right of the middle point, we get 4 and further count 3 units from 4 to the right, we obtain 7.
Again, starting from 0 and count 4 units to the left of it, we obtain - 4 and further count 1 unit from its left, we get -5.

10 + (-4) = 10 - 4 = +6 or 6  [\(\because\) difference = 6, higher integer sign is' +']

We have, -3 and 4
In this case, firstly we go to (-3) and then move (4) step to the right of (-3).
Thus, we reached to (1), i.e. (-3) + (4) = 1

Number line with all numbers is shown below:

We know that a place whose temperature is highest, will be the hottest place and the place whose temperature is lowest, will be the coldest place.
Here, the temperature of hottest place (Bangalore) = 22° C and temperature of coldest place (Lahulspiti) = - 8° C
\(\therefore\) Temperature difference between the hottest and coldest places = Temperature of Bangalore - Temperature of Lahulspiti.
= 22 - (- 8) = 22 + 8 = 30
Hence, the temperature difference between the hottest and the coldest places is 30°C.

Number line with all numbers is shown below:

From the above number line, the temperature of the indicated places are as follow:

(a) Since profit of Rs 8 is earned on a bag of white cement and a loss of Rs 5 on a bag of grey cement.
Number of white cement bags sold = 3000
Number of grey cement bags sold = 5000
\(\therefore\) Profit = Total number of white cement bags X Profit per bag
= 3000 \(\times\) 8 = Rs 24000
Loss =Total number of grey cement bags X Loss per bag
= 5000 \(\times\) 5 = Rs 25000
\(\because\) Loss > Profit, so Loss = Rs 25000 - Rs 24000 = Rs 1000
(b) Let x be the number of white cement bags sold,
According to the question,
Number of white cement bags X Profit per bag = Number of grey cement bags X Loss per bag
\(\Rightarrow\) x \(\times\) 8 = 6400 \(\times\) 5 \(\Rightarrow\) x = 6400 \(\times\) 5 / 8
\(\Rightarrow\) x = 800 \(\times\) 5 = 4000
Hence, 4000 bags of white cement must be sold.

Let the number of steps moved down to be represented by negative integers and number of steps moved up to be represented by positive integers. Given that, the total number of steps to be moved down to reach water level= 9
(i) Number of steps moved down in one jump = - 3
and number of steps moved up in one jump = 2

From the above figure, monkey will reach the water level in 11 jumps.
Alternate Method:
The monkey is sitting on the topmost step i.e. the first step. While going down, the monkey jumps 3 steps down and then jumps back 2 steps up. To reach the water level, he has to jump as follows:
-3 + 2 -3 + 2 -3 + 2 -3 + 2 -3 + 2 -3 = -8
Hence, he takes 11 jumps to reach the water level.
ii) In one jump number of steps moved up = 4
In one jump, number of steps moved down = - 2

Distance covered in single jump = 4 - 2 =2
Now, total number of steps for top = 9
From the above figure, he will reach back the top step in 5 jumps.
Alternate Method
Afterdrinkingwaters,he has to jump as followsto go back:
4 -2 + 4 - 2 + 4 = 8
Hence, he takes 5 jumps to reach back the top
(iii) (a) We have,
-3 + 2 -3 + 2 - 3 + 2 - 3 + 2 - 3 + 2 -3 = -8
(b) 4 - 2 + 4 - 2 + 4 = 8
According to the question, the sum 8 in part (b) represents going up 8 steps.