Download TOPIC#18 Algebra and Graphs – Inequality and Linear Programming

TOPIC#18
Algebra and Graphs – Inequality and Linear Programming
Q1.
Solve the inequality 4k – 1 < 14 + k .
Ans.
k <5
Q2.
Solve the inequality 5t – 4 < 28 + t
Ans.
t <8
Q3.
(a)
Write down all the integer values of n for which -2 ≤ n < 4.
(b)
Find the least integer value of x for which 3 – x < 2 .
Ans.
(a)
-2, -1, 0, 1, 2, 3
Q4.
(a)
Solve the inequality -5 < 2x + 3 < 1.
(b)
Writ e down the largest integer, x , which satisfies -5 < 2x + 3 < 1.
Ans.
(a)
-4 < x < -1
Q5.
(a)
Solve the inequality -10 < 3x – 7 < -1.
(b)
Write down the largest integer, x , which satisfies -10 < 3x – 7 < -1.
Ans.
(a)
-1 < x < 2
Q6.
The distance travelled by a car was 280 km, correct to the nearest 10 km.
(b)
(b)
(b)
2
-2
c = 10
1
The time taken was 5 hours, correct to the nearest hour.
(a)
Ans.
Complete the statements in the answer space.
___________ km ≤
distance < ____________ km
___________h
time
≤
< ____________h
(b)
What was the least possible average speed for the journey?
(a)
275 km ≤ distance < 285 km
4.5 h ≤ time < 5.5 h
(b)
275 2750

 50 km/h
5.5
55
Q7.
Solve the inequality a  2  4(5  a).
Ans.
a  2  4(5  a)
Compiled By : Sir Rashid Qureshi
a  2  20  4a
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TOPIC#18
Algebra and Graphs – Inequality and Linear Programming
5a  22
a  4.4
(a)
Solve the inequality 7  y  9.
(b)
Write down the least integer value of z for which z > - 4.
Ans.
(a)
y >5
Q9.
(a)
Solve 8  2x  3  11.
(b)
Write down the greatest and least integers which satisfy 8  2x  3  11.
Ans.
(a)
5 12  x  4
Q10.
In the diagram, A is the point (-4, -1) and AC is parallel to the x - axis.
Q8.
(a)
(b)
(b)
-3
greatest = 3
least = -5
The equation of BC is y + 2x = 4.
Find the x - coordinate of C.
Ans.
(b)
The equation of AB is y = x + 3.
(a)
at C , y = -1, put it in equation of BC
y + 2x = 4
-1 + 2x = 4,
(b)
Q11.
y ≤ x + 3,
or
y ≥ -1,
x
5
2
y + 2x ≤ 4
Find all the integers which satisfy which satisfy both
2 x  7  3 and x  4
Ans.
x ∈ {-4,
Q12.
(a)
-3}
The set A  {x :1  x  3} is shown on the number line above.
(i)
Set B is shown on the number line below.
Compiled By : Sir Rashid Qureshi
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TOPIC#18
Algebra and Graphs – Inequality and Linear Programming
Complete the description.
(ii)
B  {x :........... x .............}
The set C  {x : x  3}.
Illustrate the set C using the number line below.
(b)
X  {1, 3, 5},
Y  {3, 5},
Z  {( x, y)} : x  X , y  Y , x  y)
List the members of Z.
Ans.
(a)
(i)
B = { x : -1 < x ≤ 4}
(ii)
Q13.
(b)
Z = {(1, 3), (1, 5), (3, 5) (5, 3)}
(a)
In the diagram, the unshaded region,
R, is defined by three inequalities.
Two of these are
y  2 x  2 and
y  5 x
Write down the third inequality.
(b)
Find the integer values of x which
satisfy the following.
4  2x 13  9
Ans.
(a)
2y ≥ 𝑥
(b)
x = -3 , -4
Compiled By : Sir Rashid Qureshi
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TOPIC#18
Algebra and Graphs – Inequality and Linear Programming
Q14.
(a)
(i)
Find the smallest integer k which satisfies 7k  36.
(ii)
Find the largest integer n which satisfies 3n – 1 < 26.
(b)
The diagram shows the graphs x = 1, y = 3 and y = x - 1.
The region, R, is defined by the inequalities x >1, y < 3 and y > x - 1.
Given that the point (x, y) is in the region R, Find the integer values of x and y.
Ans.
smallest k = 6
(i)
(b)
x = 2
(a)
Solve
(b)
Write down all the integers which satisfy 7  3 x  4  2.
Ans.
(a)
-1≤x<2
Q16.
The line
Q15.
(ii)
largest n = 8
(a)
y = 2
7  3 x  4  2.
(b)
- 1, 0, 1
is drawn on the grid below.
(a)
Write down the equation of the line
(b)
On the grid,
Compiled By : Sir Rashid Qureshi
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TOPIC#18
Algebra and Graphs – Inequality and Linear Programming
(i)
Draw and label the lines x = 1,
y = 3 and x + y = 2,
(ii)
Shade the region which satisfies the three inequalities.
x  1, y  3 and x  y  2.
Ans.
(a)
y = 2x + 3
(b)
(i), (ii)
Diagram
Q17.
In the diagram, A is the point (6, 3) and C is the point (-8, -4).
The equation of AB is y = 3 and the equation of CB is y = x + 4.
(a)
Find the coordinates of B .
(b)
The unshaded region R inside triangle ABC is defined by three inequalities.
One of these is y < x + 4.
Write down the other two inequalities.
B (-1, 3)
and
y  12 x
(a)
Q18.
The shaded region inside the triangle ABC is defined by three inequalities.
Compiled By : Sir Rashid Qureshi
(b)
y <3
Ans.
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TOPIC#18
Algebra and Graphs – Inequality and Linear Programming
1
2
One of these is x  y  5 .
Ans.
(a)
Write down the other two inequalities.
(b)
How many points, with integer coordinates, lie in the shaded region?
(a)
y < 2x
(b)
Points with integer coordinates = 3
and
y >1
Compiled By : Sir Rashid Qureshi
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