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Maths

7 7 Linear equations and inequalities 7
Maths as language
Exercises
x − 3 = 5 x minus 3 is equal to 5
− 3 = 5
x x
1 Solve the equations. Solve the equations. 2x + 3 = 11 1 2 2 2x plus 3 is equal to 11
x
+ 3 = 1
x x
(a) 7x − 4 = 2x + 21 (b) 9x = 24 − x (c) 3x − 18 = −16 (d) 10x − 7 = 3x + 28 −32x = −4 minus 32 2 x
x
= −4
x
min
u
s 3
minus 32x is equal to minus 4
2(7x − 3) = 5
2(7 x x x − 3) = 5 2 t 2 t 2 times 7x minus 3 is equal to 5
2(7
x
ime
s 7
s 7
ime
2 Solve the equations. Solve the equations.
(a) 5( x (b) 3(2x + 3) = 45 (c) 7( x − 2) = 2( x + 3) (d) 4[2( x x = 15 x over 3 is equal to 15
+ 3) = 45
+ 3) = 45
(a) 5(x − 4) = −15
− 2) = 2(
(c) 7(
(c) 7(x − 2) = 2(x + 3)
(a) 5(
(d) 4[2(x − 3) + 5] = 12
+ 3)
(d) 4[2(
3
x ≤ 3 x is less than or equal to 3
x x
≤ 3
3 Solve the equations. Solve the equations. x x x < −1 x is less than minus 1 or less than negative 1
< −1
(a) x − 30 = 10 (b) −5 + x = 0 2 − x > 0
x
> 0
x
15 20 2 minus x is greater than 0
x
x
2 2
> 4
x
B − C 2x > 4 2x is greater than 4 Explorers! Maths 1, Student’s Book Explorers! Maths 1, Student’s Book
, when
C
4 Given that Given that A = B , find the value of C, when A = 10 and B = 8.
C
5 Three consecutive integers have a sum equal to 63. Form an equation and find the three integers. Three consecutive integers have a sum equal to 63. Form an equation and find the three integers.
Unit at a glance
6 Jack is Jack is y years old. His brother is 2 years older than him. The sum of their ages is 24 years.
How old is Jack?
1 A linear expression is an algebraic expression in which each term can be constant, a single variable or a
7 Solve the linear inequalities. 5 single variable multiplied by a coefficient. All variables should be raised to the power of 1.
1
(a) 2x ≥ 18 (b) 5x ≤ 135 (c) 3x − x > 8 Linear expressions: 2x + y − 5, w + 2
≥ 5(
≥ 5(
(d) 3x + 4x < 7 (e) 7x ≥ 5(x + 2) (f) 3(2 + 2x) + 5(2 − x) ≥ −8 Non-linear expressions: x 2 + 3, xy + w
x
x
x
− 4) < 20 +
value of
x
eatest possible integer
and find the gr
8 Solve the inequality 2(3x − 4) < 20 + x and find the greatest possible integer value of x. 2 A linear equation is an equation in which both sides contain only linear expressions.
x
3
Linear equations: 2x = − 5, w + = 0
x
x
x
x
x
− 2).
+ 4 > 2(
+ 4 > 2(
+ 4 > 2(
9 (a) Solve the inequality 12x + 4 > 2(x − 2). 7
z
x
z
which 3(2
x
+ 9) > 5.
x
x
(b) Find the smallest possible integer value of x for which 3(2x + 9) > 5. Non-linear equations: z 2 + 8 = zx, xy + w = 6
for
x
10 A rectangle has sides of (x A rectangle has sides of (x A rectangle has sides of ( + 1) cm and (2x + 1) cm and (2x x x x + 3) cm. 3 A fractional equation is an equation where one or more of the coefficients of the unknown variable is a
+ 3) cm.
(a) Given that the perimeter is at least 40 cm, form an inequality and show that x ≥ 5 1 . 3 fraction. Fractional equation: 1 x + 4 − x − 2 = 0
(b) If x is a prime number, 2 6
(i) find the smallest possible value of x.
(ii) find the perimeter of the rectangle for this value of x. 4 Solving a linear equation means finding the value of the unknown quantity. We can check whether our
answer is correct or not by replacing the variable in the starting equation with the value we found.
More Exercises 5 The mathematical equations that relate two or more variables and are always true are called formulas. We
say that a is the subject of a formula if a is expressed in terms of other variables. When we find the value
1 The length of a rectangle is (2y The length of a rectangle is (2y The length of a rectangle is (2 + 3) cm and the breadth is 5 cm. If its area is 50 cm 2 , find y. of the subject of a formula, we evaluate this subject.
2 The equal sides of an isosceles triangle are each equal to twice the length of its third side. If its third 6 Solving a word problem using an algebraic expression requires forming an equation and then solving it.
side is y cm long and its perimeter is 30 cm, find the length of all the sides. When we solve a word problem, we should always check whether the answer we find is acceptable.
k
3 A pencil costs k cents and a ruler costs twice as much. The total cost of 15 pencils and 6 rulers is k cents and a ruler costs twice as much. The total cost of 15 pencils and 6 rulers is 7 An inequality is a mathematical expression with two parts separated by an inequality sign. A linear
$4.05. Find the cost of a ruler and the cost of a pencil. inequality is an inequality in which both sides contain only linear expressions.
4 The heights of two bridges are (3h − 2) m and (2h − 1) m. The difference in height of the two
bridges is 5 m. What is the value of h?
is a positive whole number
om it and then multiply the r
esult by 5, the
.
When we subtract 6 fr
5 x x x is a positive whole number. When we subtract 6 from it and then multiply the result by 5, the
and find the values of
and find the values of
x
x
and find the values of
110 answer is less than 10. Form a linear inequality in terms of x and find the values of x. x 111
5 5 Algebraic expressions Review 1 - 6

1 - 6
Assessment 1 Represent the numbers on the number line.
Read the questions carefully. For each question, 4 options are given. Circle the correct one. (a) odd numbers between 16 and 25
− 2) + 5(2
t
t
+ 4).
t
t
1 Expand and simplify the expression 4(3t − 2) + 5(2t + 4).
t
t
t
t
− 5
(a) 14t + 11 1 (b) 22t + 12 (c) 3 t t − 20 (d) 6t − 5 (b) numbers smaller than 3
(d) 6
(c) 3t − 20
t
+ 1
+ 12
t
(b) 22
2 a · 0 equals . (c) numbers bigger than or equal to 10 and smaller than 20
(a) a (b) 1 (c) 0 (d) –a
− 2
− 2
= 4 and
− 2
x
y
x
x
3 Evaluate −3x − 2y + 7 when x = 4 and y = 1. Explorers! Maths 1, Student’s Book Explorers! Maths 1, Student’s Book (d) 5, 9, 13 and 18 Explorers! Maths 1, Student’s Book Explorers! Maths 1, Student’s Book
x
(a) 17 (b) 21 (c) –3 (d) –7
k
+ 3
k
4 4(−2k + 3l) is equal to . 2 Complete the sentences.
k
k
(b) –8k + 12l
+ 12
(d) 8
(a) 8k + 12l l l (b) –8 k k + 12 l l (c) –8 k k – 12 l l (d) 8k – 12l (a) The absolute value of |−47| is .
(c) –8k – 12l
+ 2 − 5
x
x
+ 3 is equal to
+ 2 − 5
5 1 + 2x − y + 2 − 5x + 3 is equal to . (b) The graph of quantities that are in inverse proportion is a line.
x
x
x
−
(a) 2 – 3x + y (b) −4 + 7x + y (c) 6 – 3x – y (d) 7x – y (c) 13,5% is equal to the decimal .
(d) The method of analysing a number into parts depending on the place values is called .
k
k
6 −k + 3l − 1 − 2k − 3l + 5 is equal to . (e) The number 8 in 3 8 is called the .
+ 3
− 3
k
k
− 1 − 2
+ 5 is equal to
l
l
l
l
(a) −k + 6 (b) 6 k k − 3 l l + 4 (c) 6 − k − 3l l l (d) –3k + 4
(b) 6k − 3l + 4
k
k
− 3
(d) –3
(c) 6 −
+ 6
k
k
Evaluate −Z − 3 + 2w − 3Z − 5w when Z = 2 and w = 0.
7 Evaluate − Z Z Z − 3 + 2 w w − 3 Z Z − 5 w w when Z Z = 2 and 3 Write the numbers in order using < or > signs.
Evaluate −
(a) 5 (b) 3 (c) −11 (d) 2
(a) Ascending order: 7, −12, 4, 0, 6
(b) Descending order: 4 , 4 , 1 , 3 , 17
10 5 2 4 20
S
=
S
S
8 Given the formula S = n(n + 1) , find the value of S when n = 25. (c) Ascending order: 3,313, 3,07, 3,309, 3,12, 3,6
S
when
2
(a) 125 (b) 225 (c) 169 (d) 325
(d) Descending order: 6 , 6 , 6 , 6 , 6 9
11
7
10
8
−
x − y
x x
x
Z
Z
= 3, find the value of
= 4,
x
9 Given that x = 4, y = 0 and Z = 3, find the value of 3Z + (xy+ (xy+ ( ZxyZxy )Z)Z 3 .
(a) 8 9 (b) 1 2 (c) 4 9 (d) − 4 9 4 Change mixed numbers into improper fractions, proper fractions into decimals and decimals into decimal
fractions or vice versa.
4
(a) 5 7 = (b) 0,096 = (c) 7 3 =
10 Factorise 5xFactorise 5xFactorise 5 − 5x − 5x − 5y − 5 y + 5Z. 10
x
+
y
(b) 5(
(b) 5(
x
(d) 5(
y
(d) 5(
(a) 5( x ) ) (b) 5(x − y − z) ) ) (c) 5( x ) ) (d) 5(x · y + z) (d) 725 = (e) 42 50 = (f) 156 =
(c) 5(
(c) 5(x − y + z)
(a) 5(
(a) 5(x · y · z)

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