Chapter 2: Linear Equations and Inequalities Lecture notes Math 1010
Section 2.1: Linear Equations
Definition of equation
An equation is a statement that equates two algebraic expressions. Solving an equation involving a variable means finding all values of the variable for which the equation is true. Such values are solutions and are said to satisfy the equation. The solution set of an equation is the set of all solutions of the equation. If an equation has the set of all real numbers as its solution set, then it is called an identity . An equation whose solution set is not the entire set of real numbers is called a conditional equation.
Ex.1
Equations.
(1) 4 x + 2 = 9
(2) 2 x − 4 = 2( x − 2)
(3) x 2 − 9 = 0
Ex.2
Checking a solution of an equation.
Determine whether x = − 3 is a solution of − 3 x − 5 = 4 x + 16 .
Properties of equalities
Two equations that have the same set of solutions are equivalent equations . For example, x = − 5 and x +5 = 0 are equivalent. An equation can be transformed into an equivalent equation using one or more of the following properties.
(1) Simplify either side : remove symbols of grouping, combine like terms, or simplify fractions on one or both sides of the equation.
(2) Apply the addition property equality: add or subtract the same quantity to/from each side of the equation.
(3) Apply the multiplication property of equality: multiply or divide each side of the equation by the same nonzero quantity.
(4) Interchange sides: interchange the two sides of the equation.
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Chapter 2: Linear Equations and Inequalities Lecture notes
Ex.3
Solve 7 = 5 x − 2 x + 1 .
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Definition of linear equation
A linear equation in one variable x is an equation that can be written in the standard form ax + b = 0 , where a and b are real numbers with a = 0 . A linear equation in one variable is also called a first-degree equation because its variable has exponent 1 . You can find the solution in the following way:
Ex.4
Solve 4 x − 12 = 0 . Then check the solution.
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Chapter 2: Linear Equations and Inequalities Lecture notes
Ex.5
Solving an equation in nonstandard form.
Solve 6( y − 1) = 4 y − 2 . Then check the solution.
Ex.6
Solve x
4
− 3
8
= 9 .
Ex.7
Solve 0 .
45 x + 1 .
2( x − 3) = 39 .
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Chapter 2: Linear Equations and Inequalities Lecture notes
Ex.8
Solve 2 x − 4 = 2( x − 3) .
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Ex.9
Solve 3 x + 2 + 2( x − 6) = 5( x − 2) .
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Chapter 2: Linear Equations and Inequalities Lecture notes
Section 2.2: Linear Equations and Problem Solving
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Ex.1
You have accepted a job at an annual salary of $40 , 830 . This salary includes a year-end bonus of $750 . You are paid twice a month. What will your gross pay be for each paycheck? Write an algebraic expression that represents this problem. Then solve the equation and answer the question.
Definition of percent
A rate is a fraction that compares two quantities measured in different units. Rates that describe increases, decreases, and discounts are often given as percents. The word percent ( = per cent) means divided by one hundred.
Ex.2
(1)
(2)
(3)
(4)
10% =
50% =
25% =
12 .
5% =
10
100
50
100
25
100
= 0
=
=
12 .
5
100
1
2
1
4
=
.
1
= 0
= 0
1
8
.
.
5
25
= 0 .
125
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Chapter 2: Linear Equations and Inequalities Lecture notes
Ex.3
The number 13 is 20% of what number?
Ex.4
The number 28 is what percent of 80 ?
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Chapter 2: Linear Equations and Inequalities Lecture notes Math 1010
Section 2.4: Linear Inequalities
Algebraic inequalities
Algebraic inequalities are inequalities that contain one or more variable terms. For example, x < 3 , x ≥ 5 and x − 2 > 2 x + 5 are algebraic inequalities. Solving an inequality in the variable x means that you need to find the values of x for which the inequality is true. Those values are the solutions of the inequality and the set of all solutions is called solution set. The graph of an inequality is the plot of the solutions in the real line.
Bounded and unbounded intervals
Let a and b be real numbers such that a < b . The following intervals are called bounded intervals:
The length of intervals [ a, b ] , [ a, b ) , ( a, b ] and ( a, b ) is b − a . If an interval doesn’t have a finite length, then it is called unbounded. The following intervals are unbounded:
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Chapter 2: Linear Equations and Inequalities Lecture notes
Ex.1
Sketch the graph of each inequality.
(1) − 5 < x < 8
(2) x ≤ − 1
(3) 4 ≤ x ≤ 6
(4) x ≥ 2
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Properties of inequalities
Let a , b , and c be real numbers, variables, or expressions.
(1) Addition and subtraction properties
Adding the same quantity to (subtracting the same quantity from) each side of an inequality produces an equivalent inequality (that is an inequality with the same solution set).
If a < b , then a + c < b + c .
If a < b , then a − c < b − c .
(2) Multiplication and division properties
Multiplying (dividing) each side of an inequality by a positive quantity produces an equivalent inequality. Multiplying (dividing) each side of an inequality by a negative quantity produces an equivalent inequality in which the inequality symbol is reversed.
If a < b , and c > 0 then ac < bc .
If a < b , and c > 0 , then a c
< b c
.
If a < b , and c < 0 then ac > bc .
If a < b , and c < 0 , then a c
> b c
.
(3) Transitive property
If a < b and b < c , then a < c .
Linear inequalities
An inequality is called linear if it has one of the following forms: ax + b ≤ 0 , ax + b < 0 , ax + b ≥ 0 , ax + b > 0
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Chapter 2: Linear Equations and Inequalities Lecture notes
Ex.2
Solve x + 4 < 9 .
Ex.3
Solve 12 − 4 x ≤ 30 .
Ex.4
Solve 7 x − 3 > 3( x + 1) .
Ex.5
Solve 2 x
3
+ 12 < x
6
+ 18 .
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Chapter 2: Linear Equations and Inequalities Lecture notes Math 1010
Compound inequalities
Two inequalities joined by the word and or the word or give a compound inequality. When two inequalities are joined by the word and , the solution set consists of all real numbers that satisfy both inequalities. When two inequalities are joined by the word or , the solution set consists of all real numbers that satisfy one of the two inequalities. A compound inequality formed by the word and is called conjunctive and a compound inequality formed by the word or is called disjunctive .
Ex.6
Solve − 7 ≤ 5 x − 2 < 8 .
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Chapter 2: Linear Equations and Inequalities Lecture notes
Ex.7
Solve the compound inequality − 1 ≤ 2 x − 3 and 2 x − 3 < 5 .
Ex.8
Solve the compound inequality − 1 ≤ 2 x − 3 or 2 x − 3 < 5 .
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Ex.9
A solution set is
(1) Write the solution set as a compound inequality.
(2) Write the solution set using the union symbol.
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Chapter 2: Linear Equations and Inequalities Lecture notes
Ex.10
Write the compound inequality − 3 ≤ x ≤ 5 using the intersection symbol.
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Ex.11
(1) x is at most n : x ≤ n .
(2) x is no more that n : x ≤ n .
(3) x is at least n : x ≥ n .
(4) x is no less than n : x ≥ n .
(5) x is more than n : x > n .
(6) x is less than n : x < n .
(7) x is a minimum of n : x ≥ n .
(8) x is at least m , but less than n : m ≤ x < n .
(9) x is greater than m , but no more than n : m < x ≤ n .
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Chapter 2: Linear Equations and Inequalities Lecture notes Math 1010
Section 2.5: Absolute Value Equations and Inequalities
Solving absolute value equation
Let x be a variable or an algebraic expression and let a be a real number such that a ≥ 0 . The solutions of the equation | x | = a are x = a and x = − a .
Ex.1
Solve each absolute value equation.
(1) | x | = 12
(2) | y | = − 3
(3) | z | = 0
(4) | x | = 3
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Chapter 2: Linear Equations and Inequalities Lecture notes
Ex.2
Solve | 3 x + 4 | = 10 .
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Ex.3
Solve | 2 x − 1 | + 3 = 8 .
Ex.4
Solve | 3 x − 4 | = | 7 x − 16 | .
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Chapter 2: Linear Equations and Inequalities Lecture notes
Ex.5
Solve | x + 5 | = | x + 11 | .
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Solving absolute value inequality
Let x be a variable or an algebraic expression and let a be a real number such that a > 0 .
(1) The solutions of | x | < a are values of x such that − a < x < a .
(2) The solutions of | x | > a are values of x such that x < − a or x > a .
Ex.6
Solve | x − 5 | < 2 .
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Chapter 2: Linear Equations and Inequalities Lecture notes
Ex.7
Solve | 3 x − 4 | ≥ 5 .
Ex.8
Solve | 2 − x
3
| < 0 .
01 .
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