Download Solving Equations Identities and Contradictions

Math 95
7.1 "Review of Equations and Inequalities"
Objectives:
*
Solving Equations
*
Identities and Contradictions
*
Solving Formulas
*
Problem Solving
*
Inequalities
*
Compound Inequalities
Solving Equations
Recall than an equation is a statement indicating that two mathematical expressions are equal. The set of numbers
that satisfy an equation is called its solution set. Finding the solution set of an equation is called solving the equation.
Solving Equations:
1: If an equation contains fractions, multiply both sides of the equation by their least common denominator (LCD).
2: Use the distributive property to remove all grouping symbols and combine like terms.
3: Use the addition and subtraction properties to get all variables on one side of the equation and
all numbers on the other side. Combine like terms, if necessary.
4: Use the multiplication and division properties to make the coe¢ cient of the variable equal to 1.
5: Check the result by replacing the variable with the possible solution and verifying that the number satis…es the equation.
Example 1: (Solving equations)
Solve the following equations.
a) 8 (3a
5)
4 (2a + 3) = 12
b)
2
(x
3
2) =
5
(x
2
1) + 3
Identities and Contradictions
The equations discussed so far have been conditional equations. For these equations, some numbers x are solutions
and others are not. An identity is an equation that is satis…ed by every number x for which both sides of the equation are
de…ned. A contradiction is an equation that has no solution.
Page: 1
Notes by Bibiana Lopez
Beginning and Intermediate Algebra by Gustafson and Frisk
7.1
Example 2: (Solving equations)
Solve the following equations.
a) 3 (x + 1)
(20 + x) = 5 (x
1)
3 (x + 4)
b)
x
2
3
3=
1 x+1
+
5
3
Solving Formulas
To solve a formula for a variable means to isolate that variable on one side of the equal (=) sign and place all other
quantities on the other side.
Example 3: (Solving formulas)
A sales clerk earns $200 per week plus a 5% commission on the value of the merchandise she sells. What dollar volume must
she sell each week to earn $250 in three successive weeks?
Problem Solving
Example 4: (Problem solving)
A 30-foot steel beam is to be cut into two pieces. The longer piece is to be 2 feet more than 3 times as long as the shorter
piece. Find the length of each piece.
Page: 2
Notes by Bibiana Lopez
Beginning and Intermediate Algebra by Gustafson and Frisk
7.1
Inequalities
Inequalities are statements indicating that two expressions are unequal and they contain one or more of the following
symbols. If a and b are real numbers, the following table shows the di¤erent types of intervals that can occur.
Kind of Interval
Inequality
Graph
Interval Notation
Open interval
a<x<b
(a; b)
Half-open interval
a
[a; b)
Closed interval
Unbounded interval
x<b
a<x
b
(a; b]
a
b
[a; b]
x
x>a
(a; 1)
x
[a; 1)
a
x<a
( 1; a)
x
( 1; a]
a
( 1; 1)
1<x<1
De…nition:
"Linear Inequalities"
A linear inequality in one variable (say, x) is any inequality that can be written in one of the following forms,
where a; b; and c represent real numbers and a 6= 0:
:
Examples
Non-examples
To solve a linear inequality means to …nd the values of its variable that make the inequality true. The set of all solutions
of an inequality is called its solution set. We will use the following properties to solve inequalities in one variable.
Addition and Subtraction Properties of Inequality:
Adding or subtracting the same number from both sides of an inequality does not change the solutions. For any real
numbers a; b; and c :
Similar statements can be made for the symbols
and
:
, > , or
Page: 3
Notes by Bibiana Lopez
Beginning and Intermediate Algebra by Gustafson and Frisk
7.1
Multiplication and Division Properties of Inequality:
Multiplying or dividing by the same positive number does not change the solutions. For any real numbers
a; b; and c (where c is positive) :
:
If we multiply or divide by a negative number, the direction of the inequality must be reversed. For any
real numbers a; b; and c (where c is negative) :
Similar statements can be made for the symbols
:
; >; or
Example 5: (Solving inequalities)
Solve the following inequalities. Graph the solution set and write it using interval notation.
1
2
a) 3 (2x 9) < 9
b)
(x 1)
(x + 1)
2
3
c) 8
(6 + 5m) >
9m
(3
4m)
d) 7
Page: 4
2 (x + 3)
4x
6 (x
3)
Notes by Bibiana Lopez
Beginning and Intermediate Algebra by Gustafson and Frisk
7.1
Solving Compound Inequalities
De…nition:
"The Intersection of Two Sets"
The intersection of set A and set B;
is the set of all elements that are common to set A and set B:
;
De…nition:
"The Union of Two Sets"
The union of set A and set B;
is the set of elements that belong to set A or set B or both.
;
Example 6: (Solving compound inequalities)
Solve the following compound inequalities. Graph the solution set and write it using interval notation.
a) 3x + 4 <
De…nition:
2 or 3x + 4 > 10
b) 5 (x
2)
0 and
3x < 9
"Double Inequalities"
kThe double inequality c < x < d is equivalent to c < x and x < dk
Example 7: (Solving double inequalities)
Solve the following double inequalities. Graph the solution set and write it using interval notation..
a)
3
2x + 5 < 7
b) 25 > 3x
Page: 5
2
7
Notes by Bibiana Lopez