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Page 1
chaPTer 5 EqUATIONS AND INEqUALITIES 129
5
equations and inequalities
In this chapter, you will:
• revise solving linear equations and quadratic equations
by factorisation
• solve simultaneous linear equations with two variables
• revise notation (interval, set builder, number line, sets)
• solve simple linear inequalities (and show solutions
graphically)
• manipulate formulae (technical related) and solve word
problems involving linear, quadratic, or simultaneous
linear equations
2153 TechMaths Eng G10 LB.indb 129 2015/10/22 3:40 PM
Page 2
chaPTer 5 EqUATIONS AND INEqUALITIES 129
5
equations and inequalities
In this chapter, you will:
• revise solving linear equations and quadratic equations
by factorisation
• solve simultaneous linear equations with two variables
• revise notation (interval, set builder, number line, sets)
• solve simple linear inequalities (and show solutions
graphically)
• manipulate formulae (technical related) and solve word
problems involving linear, quadratic, or simultaneous
linear equations
2153 TechMaths Eng G10 LB.indb 129 2015/10/22 3:40 PM
130 Technical Ma TheMaTicS Grade 10
5.1 Getting ready to learn
w orking with number relationships
When we work with variables, we need to remember our ways of working with numbers. We use
exactly the same rules for variables.
w orked example
In each of the examples below, the number sentences have been rearranged. Read through
these examples carefully and discuss:
A. What have we done to each equation to change the way it is written?
B. Do the equations stay true in each case?
8 + 5 = 13
8 = 13 - 5
5 = 13 - 8
6 × 4 = 24
6 =
24
___
4
4 =
24
___
6
9 × 3 + 7 = 34
9 × 3 = 34 - 7
9 =
34 - 7
______
3
3 =
34 - 7
______
9
5.2 Working with equations
An equation in x means an equation that contains the variable x. This is an invitation to
calculate all possible values of x so that two given expressions in x on either side of the equals
sign are equal. Any number that we substitute for x that makes the equality true is called a
solution.
linear equations in one variable
To solve an equation in one variable, we pretend to know what the solution is. Then we arrange
the equation to calculate what the variable is.
Here is an example using letters a, b, and c for the known numbers:
x + b = c
x = c - b
ax + b = c
ax = c - b
x =
c - b
_____
a
ax = b
x =
b
__
a
2153 TechMaths Eng G10 LB.indb 130 2015/10/22 3:40 PM
Page 3
chaPTer 5 EqUATIONS AND INEqUALITIES 129
5
equations and inequalities
In this chapter, you will:
• revise solving linear equations and quadratic equations
by factorisation
• solve simultaneous linear equations with two variables
• revise notation (interval, set builder, number line, sets)
• solve simple linear inequalities (and show solutions
graphically)
• manipulate formulae (technical related) and solve word
problems involving linear, quadratic, or simultaneous
linear equations
2153 TechMaths Eng G10 LB.indb 129 2015/10/22 3:40 PM
130 Technical Ma TheMaTicS Grade 10
5.1 Getting ready to learn
w orking with number relationships
When we work with variables, we need to remember our ways of working with numbers. We use
exactly the same rules for variables.
w orked example
In each of the examples below, the number sentences have been rearranged. Read through
these examples carefully and discuss:
A. What have we done to each equation to change the way it is written?
B. Do the equations stay true in each case?
8 + 5 = 13
8 = 13 - 5
5 = 13 - 8
6 × 4 = 24
6 =
24
___
4
4 =
24
___
6
9 × 3 + 7 = 34
9 × 3 = 34 - 7
9 =
34 - 7
______
3
3 =
34 - 7
______
9
5.2 Working with equations
An equation in x means an equation that contains the variable x. This is an invitation to
calculate all possible values of x so that two given expressions in x on either side of the equals
sign are equal. Any number that we substitute for x that makes the equality true is called a
solution.
linear equations in one variable
To solve an equation in one variable, we pretend to know what the solution is. Then we arrange
the equation to calculate what the variable is.
Here is an example using letters a, b, and c for the known numbers:
x + b = c
x = c - b
ax + b = c
ax = c - b
x =
c - b
_____
a
ax = b
x =
b
__
a
2153 TechMaths Eng G10 LB.indb 130 2015/10/22 3:40 PM
chaPTer 5 EqUATIONS AND INEqUALITIES 131
w orked examples
Solve for x. Check your answer in each case.
A. Problem: x + 4 = 7
Solution: x = 7 - 4
x = 3 Check: 3 + 4 = 7
B. Problem: 2x = 5
Solution: x =
5
__
2
Check: 2
(
?
5
__
2
)
= 5
x = 2
1
__
2
C. Problem: 2x + 5 = 1 1
Solution: 2x = 1 1 - 5
x =
11 - 5
______
2
x =
6
__
2
x = 3 Check: 2 × 3 + 5 = 6 + 5 = 1 1
exercises
1 Solve for x and check your answer in each case.
(a) x + 2 = 0 (b) x + 2 = -4 (c) 2x = 0
(d) 2x = -4 (e)
1
__
3
x = 9 (f) x - 2 = -4
2 Solve for x in each of the following:
(a) 4x + 1 = 9 (b) 4x + 1 = -7 (c) 4x + 1 = 2
(d) 4x + 1 = 0 (e) 4x + 1 = -2 (f) 4x + 1 = 4
(g) 4x + 1 = 1 (h) 4x + 1 = -1 1
More equations
Linear equations of the form Ax + B = Cx + D; (A, B, C, and D are constants, and C ? A.)
We assume that x is a number, so that Ax + B = Cx + D, then:
Ax - Cx = D - B
(A - C)x = D - B
x =
D - B
_____
A - C
2153 TechMaths Eng G10 LB.indb 131 2015/10/22 3:40 PM
Page 4
chaPTer 5 EqUATIONS AND INEqUALITIES 129
5
equations and inequalities
In this chapter, you will:
• revise solving linear equations and quadratic equations
by factorisation
• solve simultaneous linear equations with two variables
• revise notation (interval, set builder, number line, sets)
• solve simple linear inequalities (and show solutions
graphically)
• manipulate formulae (technical related) and solve word
problems involving linear, quadratic, or simultaneous
linear equations
2153 TechMaths Eng G10 LB.indb 129 2015/10/22 3:40 PM
130 Technical Ma TheMaTicS Grade 10
5.1 Getting ready to learn
w orking with number relationships
When we work with variables, we need to remember our ways of working with numbers. We use
exactly the same rules for variables.
w orked example
In each of the examples below, the number sentences have been rearranged. Read through
these examples carefully and discuss:
A. What have we done to each equation to change the way it is written?
B. Do the equations stay true in each case?
8 + 5 = 13
8 = 13 - 5
5 = 13 - 8
6 × 4 = 24
6 =
24
___
4
4 =
24
___
6
9 × 3 + 7 = 34
9 × 3 = 34 - 7
9 =
34 - 7
______
3
3 =
34 - 7
______
9
5.2 Working with equations
An equation in x means an equation that contains the variable x. This is an invitation to
calculate all possible values of x so that two given expressions in x on either side of the equals
sign are equal. Any number that we substitute for x that makes the equality true is called a
solution.
linear equations in one variable
To solve an equation in one variable, we pretend to know what the solution is. Then we arrange
the equation to calculate what the variable is.
Here is an example using letters a, b, and c for the known numbers:
x + b = c
x = c - b
ax + b = c
ax = c - b
x =
c - b
_____
a
ax = b
x =
b
__
a
2153 TechMaths Eng G10 LB.indb 130 2015/10/22 3:40 PM
chaPTer 5 EqUATIONS AND INEqUALITIES 131
w orked examples
Solve for x. Check your answer in each case.
A. Problem: x + 4 = 7
Solution: x = 7 - 4
x = 3 Check: 3 + 4 = 7
B. Problem: 2x = 5
Solution: x =
5
__
2
Check: 2
(
?
5
__
2
)
= 5
x = 2
1
__
2
C. Problem: 2x + 5 = 1 1
Solution: 2x = 1 1 - 5
x =
11 - 5
______
2
x =
6
__
2
x = 3 Check: 2 × 3 + 5 = 6 + 5 = 1 1
exercises
1 Solve for x and check your answer in each case.
(a) x + 2 = 0 (b) x + 2 = -4 (c) 2x = 0
(d) 2x = -4 (e)
1
__
3
x = 9 (f) x - 2 = -4
2 Solve for x in each of the following:
(a) 4x + 1 = 9 (b) 4x + 1 = -7 (c) 4x + 1 = 2
(d) 4x + 1 = 0 (e) 4x + 1 = -2 (f) 4x + 1 = 4
(g) 4x + 1 = 1 (h) 4x + 1 = -1 1
More equations
Linear equations of the form Ax + B = Cx + D; (A, B, C, and D are constants, and C ? A.)
We assume that x is a number, so that Ax + B = Cx + D, then:
Ax - Cx = D - B
(A - C)x = D - B
x =
D - B
_____
A - C
2153 TechMaths Eng G10 LB.indb 131 2015/10/22 3:40 PM
132 Technical Ma TheMaTicS Grade 10
w orked examples
A. Problem: Solve for x: 3x + 4 = 2x + 1
Solution: 3x + 4 = 2x + 1
3x - 2x = 1 - 4
(3 - 2)x = 1 - 4
x =
1 - 4
_____
3 - 2
x =
- 3
____
1
= - 3 Check: 3( - 3) + 4 = - 9 + 4 = - 5, and 2(- 3) + 1 = - 6 + 1 = - 5
So x = -3 is the solution of 3x + 4 = 2x + 1
B. Problem: Solve for x:
x
__
5
+ 1 = 2x - 8
S olu tion:
x
__
5
+ 1 = 2x - 8
x
__
5
- 2x = - 8 - 1
(
?
1
__
5
- 2
)
x = - 8 - 1
x =
- 8 - 1
_______
1
__
5
- 2
x =
- 9
___
- 9
___
5
x = 5 Check: x =
5
__
5
+ 1 = 2, and 2(5) - 2 = 1 0 - 8 = 2
So x = 5 is the solution to
x
__
5
+ 1 = 2x - 8
exercise
3 Solve for x in each of the following:
(a) 5x = 3x + 2 (b) 5x - 1 = 3x + 13
(c) -7x + 8 = 2 - 13x (d) x + 3 = 2x - 4
(e)
x
__
3
+ 4 = 2x - 1 (f)
x
__
3
- 7 = 9 - x
(g)
x
__
2
+ 10 = 85 - 7x (h) 5 +
x
__
4
= -2x +
1
__
2
2153 TechMaths Eng G10 LB.indb 132 2015/10/22 3:40 PM
Page 5
chaPTer 5 EqUATIONS AND INEqUALITIES 129
5
equations and inequalities
In this chapter, you will:
• revise solving linear equations and quadratic equations
by factorisation
• solve simultaneous linear equations with two variables
• revise notation (interval, set builder, number line, sets)
• solve simple linear inequalities (and show solutions
graphically)
• manipulate formulae (technical related) and solve word
problems involving linear, quadratic, or simultaneous
linear equations
2153 TechMaths Eng G10 LB.indb 129 2015/10/22 3:40 PM
130 Technical Ma TheMaTicS Grade 10
5.1 Getting ready to learn
w orking with number relationships
When we work with variables, we need to remember our ways of working with numbers. We use
exactly the same rules for variables.
w orked example
In each of the examples below, the number sentences have been rearranged. Read through
these examples carefully and discuss:
A. What have we done to each equation to change the way it is written?
B. Do the equations stay true in each case?
8 + 5 = 13
8 = 13 - 5
5 = 13 - 8
6 × 4 = 24
6 =
24
___
4
4 =
24
___
6
9 × 3 + 7 = 34
9 × 3 = 34 - 7
9 =
34 - 7
______
3
3 =
34 - 7
______
9
5.2 Working with equations
An equation in x means an equation that contains the variable x. This is an invitation to
calculate all possible values of x so that two given expressions in x on either side of the equals
sign are equal. Any number that we substitute for x that makes the equality true is called a
solution.
linear equations in one variable
To solve an equation in one variable, we pretend to know what the solution is. Then we arrange
the equation to calculate what the variable is.
Here is an example using letters a, b, and c for the known numbers:
x + b = c
x = c - b
ax + b = c
ax = c - b
x =
c - b
_____
a
ax = b
x =
b
__
a
2153 TechMaths Eng G10 LB.indb 130 2015/10/22 3:40 PM
chaPTer 5 EqUATIONS AND INEqUALITIES 131
w orked examples
Solve for x. Check your answer in each case.
A. Problem: x + 4 = 7
Solution: x = 7 - 4
x = 3 Check: 3 + 4 = 7
B. Problem: 2x = 5
Solution: x =
5
__
2
Check: 2
(
?
5
__
2
)
= 5
x = 2
1
__
2
C. Problem: 2x + 5 = 1 1
Solution: 2x = 1 1 - 5
x =
11 - 5
______
2
x =
6
__
2
x = 3 Check: 2 × 3 + 5 = 6 + 5 = 1 1
exercises
1 Solve for x and check your answer in each case.
(a) x + 2 = 0 (b) x + 2 = -4 (c) 2x = 0
(d) 2x = -4 (e)
1
__
3
x = 9 (f) x - 2 = -4
2 Solve for x in each of the following:
(a) 4x + 1 = 9 (b) 4x + 1 = -7 (c) 4x + 1 = 2
(d) 4x + 1 = 0 (e) 4x + 1 = -2 (f) 4x + 1 = 4
(g) 4x + 1 = 1 (h) 4x + 1 = -1 1
More equations
Linear equations of the form Ax + B = Cx + D; (A, B, C, and D are constants, and C ? A.)
We assume that x is a number, so that Ax + B = Cx + D, then:
Ax - Cx = D - B
(A - C)x = D - B
x =
D - B
_____
A - C
2153 TechMaths Eng G10 LB.indb 131 2015/10/22 3:40 PM
132 Technical Ma TheMaTicS Grade 10
w orked examples
A. Problem: Solve for x: 3x + 4 = 2x + 1
Solution: 3x + 4 = 2x + 1
3x - 2x = 1 - 4
(3 - 2)x = 1 - 4
x =
1 - 4
_____
3 - 2
x =
- 3
____
1
= - 3 Check: 3( - 3) + 4 = - 9 + 4 = - 5, and 2(- 3) + 1 = - 6 + 1 = - 5
So x = -3 is the solution of 3x + 4 = 2x + 1
B. Problem: Solve for x:
x
__
5
+ 1 = 2x - 8
S olu tion:
x
__
5
+ 1 = 2x - 8
x
__
5
- 2x = - 8 - 1
(
?
1
__
5
- 2
)
x = - 8 - 1
x =
- 8 - 1
_______
1
__
5
- 2
x =
- 9
___
- 9
___
5
x = 5 Check: x =
5
__
5
+ 1 = 2, and 2(5) - 2 = 1 0 - 8 = 2
So x = 5 is the solution to
x
__
5
+ 1 = 2x - 8
exercise
3 Solve for x in each of the following:
(a) 5x = 3x + 2 (b) 5x - 1 = 3x + 13
(c) -7x + 8 = 2 - 13x (d) x + 3 = 2x - 4
(e)
x
__
3
+ 4 = 2x - 1 (f)
x
__
3
- 7 = 9 - x
(g)
x
__
2
+ 10 = 85 - 7x (h) 5 +
x
__
4
= -2x +
1
__
2
2153 TechMaths Eng G10 LB.indb 132 2015/10/22 3:40 PM
chaPTer 5 EqUATIONS AND INEqUALITIES 133
solving equations with fractions
Sometimes, a linear expression disguises itself as a rational expression such as
x - 3
______
3x + 1
= 2. In cases
such as this one, we need to ensure that we are not dividing by 0, by considering only those
values for which the denominator is not 0. In other words, we need to put a restriction on the
denominator, such that it is not equal to 0.
w orked example
Problem: Solve for x in
x - 3
______
3x + 1
= 2, where x ? Q.
Solution:
x - 3
______
3x + 1
= 2, restriction for the equation to be true 3x + 1 ? 0, ? x ? -
1
__
3
x - 3 = 2(3x + 1)
x - 3 = 6x + 2
-5x = 5
x = -1 Check:
-1 - 3
_________
3(-1) + 1
=
- 4
___
- 2
= 2
Also: -1 ? -
1
__
3
(r es tr iction)
? x = -1 is the solution to
x - 3
______
3x + 1
= 2
exercise
4 Solve for x, where x ? Q (state the restriction on the denominator).
(a)
x + 1
_____
2
= 6 (b)
2
_____
x + 1
= 6
(c)
3x + 1
______
x
= 2 (d)
5x - 2
______
6x + 3
= 3
(e)
9x - 3
______
x - 1
= 3 (f)
x - 1
_____
x + 1
= -2
5.3 Simultaneous equations
linear equations in two variables
We say that a pair of linear equations, such as 2x + y = 5 and 3x + 2y = 4, is a system of linear
equations or simultaneous equations in x and y.
To solve simultaneous equations, we need to find all the ordered pairs of numbers that are
solutions of both equations. Such an ordered pair is called the solution of the system.
2153 TechMaths Eng G10 LB.indb 133 2015/10/22 3:40 PM
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