Download 1 Linear equations and inequalities

Business Mathematics I - LI, Lecture 5, October 30, 2013
1
Linear equations and inequalities
1.1
Linear equations
An algebraic equation is a mathematical statement that relates two algebraic expressions involving at least one
variable. The replacement set or definition domain for a variable is the set of numbers that are permitted to
replace the variable.
Examples: 3x − 7 = x + 5, x2 − 4x + 4 = x4 + 7x + 12 (x + 3)(x + 4) = 0, . . .
• Addition property: if x = y then x + z = y + z,
• Subtraction property: if x = y then x − z = y − z,
• Multiplication property: if x = y then xz = yz, z ̸= 0,
• Division property: if x = y then
x
z
= yz , z ̸= 0,
• Substitution property: if x = y then either may replace the other in any statement
Linear equation in one variable can be written in the standard form ax + b = 0, a ̸= 0, where a and b are real
numbers and x is variable. Its solution solution set is {− ab } or we write that x = − ab .
Example: 4x − 2 = 5(x + 1) is a linear equation which can be written in the form x + 7 = 0 and its solution is
x = −7. The equation 2x − 1 = 2x + 5 has no solution.
Graphical representation: straight line y = 3x + 1
1.2
Systems of linear equations
System of two linear equations in two variables can be written in the form ax + by = f , cx + dy = g, where x
and y are variables and a, b, c, d, f and g are real numbers. Its solution solution set is the set of all such pairs of
numbers x = x0 and y = y0 if each equation is satisfied by the pair.
Example: 3x + 2y = 12
4x − y = 5
Solving a system by substitution:
3x + 2y = 12
4x − y = 5 =⇒ y = 4x − 5
3x + 2(4x − 5) = 12
3x + 8x − 10 = 12
11x = 22
x=2
Solving a system by elimination (addition):
3x + y = 14
4x − y = 7
3x + y + 4x − y = 14 + 7
3x + 4x + y − y = 21
7x = 21
x=3
3x + y = 14
3 × 3 + y = 14
y = 14 − 9 = 5
Graphical representation: solve the system x + y = 5 and 2x − y = 1
1.3
Linear inequalities
Example: 3(x + 2) ≤ 12(x + 8), 3 ≤ 4x − 8 < 5, . . .
• Transitive property: if x < y and y < z, then x < z,
• Addition property: if x < y then x + z < y + z,
• Subtraction property: if x < y then x − z < y − z,
• Multiplication property: if x < y and z > 0 then xz < yz,
• Multiplication property: if x < y and z < 0 then xz > yz,
• Division property: if x < y and z > 0 then
x
z
< yz ,
• Division property: if x < y and z < 0 then
x
z
> yz .
Solving linear inequality:
2(3x + 2) − 14 < 8(x − 1)
6x + 4 − 14 < 8x − 8
6x − 10 < 8x − 8
6x − 10 + 8 < 8x − 8 + 8
6x − 2 < 8x
6x − 2 − 6x < 8x − 6x
−2 < 2x
− 22 < 2x
2
−1 < x
Solving double inequality:
2 ≤ 3x + 1 < 8
2 − 1 ≤ 3x + 1 − 1 < 8 − 1
1 ≤ 3x < 7
1
3x
7
3 ≤ 3 < 3
1
7
3 ≤x< 3
x ∈ [ 13 , 73 )
1.4
Rational inequalities leading to linear inequalities
Example:
2x
x+2
>1
• x + 2 > 0 =⇒ x > −2 . . . 2x > x + 2 =⇒ x > 2
• x + 2 < 0 =⇒ x < −2 . . . 2x < x + 2 =⇒ x < 2
Alternate solution:
2x
x+2 > 1
2x
x+2 − 1 > 0
x−2
x+2 > 0
• x − 2 > 0 and x + 2 > 0 ⇐⇒ x > 2 and x > −2 =⇒ x > 2
• x − 2 < 0 and x + 2 < 0 ⇐⇒ x < 2 and x < −2 =⇒ x < −2
x ∈ (−∞, −2) ∪ (2, ∞)