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1.2 Integers (Directed Numbers)

Comprehensive Notes

Essential rules for working with directed numbers.

Directed Numbers
Integers that have a direction (positive or negative). Zero is neither positive nor negative.
Rule of Signs
(+)×(+)=(+), (−)×(−)=(+), (+)×(−)=(−). Same rules apply for division.
Ordering
On a number line, numbers further to the right are greater. −2>−7 because -2 is further right.
Temperature
'Rise' means add, 'Fall' or 'Decrease' means subtract. Freezing point of water is 0°C.
BIDMAS
Brackets, Indices, Division/Multiplication, Addition/Subtraction. Always follow this order with directed numbers.

Past Paper Style Practice

Question 1: Number Line & Inequalities

(a) Mark the points 5,−2,−7,0,3 on a number line. [2]
(b) Write these integers in ascending order. [1]
(c) Find the integer exactly halfway between −7 and 3. [2]
(d) Write an inequality for all integers x such that −7≤x<3. [1]
(e) How many integers satisfy the inequality in (d)? [1]

Question 2: Temperature Change

At 06:00 the temperature is −4°C. By 12:00 it increases by 9°C. From 12:00 to 18:00 it decreases by 13°C.
(a) Find the temperature at 12:00. [1]
(b) Find the temperature at 18:00. [1]
(c) Find the total change in temperature from 06:00 to 18:00. [2]
(d) At 20:00 the temperature is −6°C. Find the change from 18:00 to 20:00. [2]

Question 3: Operations & Brackets

Evaluate the following:
(a) −7+12−19 [1]
(b) 18−(−6)−13 [1]
(c) −4(7−11) [1]
(d) (−3)2−32 [2]
(e) 48÷(−6)+(−5) [2]

Question 4: Find the Missing Integer

Find the integer for each part:
(a) −8+x=13 [1]
(b) y−(−17)=−5 [1]
(c) −6k=54 [1]
(d) m/(−7)=−9 [1]
(e) State whether each of x,y,k,m is positive, negative or zero. [2]

Question 5: Comparisons

For each pair, write the **greater** integer.
(a) −9 and −4
(b) −2 and 1
(c) 0 and −1
(d) −12 and −15
(e) Explain briefly why −4 is greater than −9 using a number line idea. [2]

Question 6: Shopping Account Context

A customer has a balance of -₹350 (they owe money). They pay ₹500, then buy items costing ₹120 and ₹90, then receive a refund of ₹75.
(a) Find the balance after paying ₹500. [1]
(b) Find the balance after buying the two items. [2]
(c) Find the final balance after the refund. [1]
(d) State whether the customer is in credit or debt at the end. [1]

Question 7: Algebraic Reasoning

An expression is E=−3x+5, where x is an integer.
(a) Find E when x=−4. [2]
(b) Find the integer x when E=20. [2]
(c) Find the integer x when E=−10. [2]
(d) Explain why E can never be equal to 4 for any integer x. [2]

Question 8: Lift/Elevator Rules

A lift starts at floor +3. It goes down 11 floors, then up 6 floors, then down 9 floors.
(a) What floor does it reach after going down 11 floors? [1]
(b) What floor does it reach after going up 6 floors? [1]
(c) What is the final floor? [1]
(d) What is the total number of floors travelled? [2]
(e) Write the final floor as a single calculation. [1]

Question 9: Reverse Operations

You are told that A=(−12)−(x)+(−5)=−30 where x is an integer.
(a) Solve for x. [2]
(b) Check your answer by substitution. [1]
(c) Write A in the form A=−17−x. [1]
(d) If A≤−30, list all integer solutions for x between −5 and 5. [3]

Question 10: Range Challenge

Let x be an integer and define T=2x−7.
(a) Find T when x=−3,x=0, and x=4. [3]
(b) Solve 2x−7=9. [2]
(c) Solve 2x−7<−1 and list the integers x from -4 to 4 that satisfy it. [3]
(d) Find all integer values of x such that −10≤2x−7≤2. [4]