Download Chapter 2: Linear Equations, Inequali

Math 40, Jagodina
Lecture Notes
Chapter 2
Chapter 2: Linear Equations, Inequalities, and Applications
Name:
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Linear Equations in One Variable
Solving a Linear Equation in One Variable
1. Clear fractions. Multiply each side of the equation by the least common denominator.
2. Simplify each side separately. Use the distributive property to clear parentheses and combine
like terms as needed.
3. Isolate the variable terms on one side. Use the addition property to get all terms with
variables on one side of the equation and all numbers on the other.
4. Isolate the variable. Use the multiplication property to get an equation with just the variable
on one side.
5. Check. Substitute the proposed solution into the original equation.
Example. Solve each equation.
1.
k+1 k+3
1
+
=
2
4
2
2. 0.02(60) + 0.04p = 0.03(50 + p)
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Lecture Notes
Chapter 2
Types of Equations
Type of Linear Equation
Conditional
Identity
Contradiction
Number of Solution
One
Infinite
All real solutions
None
No solution
Example. Solve the equation. Decide whether it is a conditional equation, and identity, or a contradiction.
7[2 − (3 + 4x)] − 2x = −9 + 2(1 − 15x)
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Formulas
Solving for a Specified Variable
Example. Solve for the specified variable.
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2. y = (x + 3) for x
2
1. m = 2k + 3b for k
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Lecture Notes
Chapter 2
Example. Distance formula. The distance is 500 mi and the rate is 25 mph. Find the time.
Example. Solving Percent Problems.
Partial amount
= Percent (represented as decimal)
Whole amount
1. A mixture of gasoline and oil contains 20 oz, of which 1 oz is oil. What percent of the mixture is
oil?
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Lecture Notes
Chapter 2
Applications of Linear Equations
Addition
sum
more than
plus
added to
increased by
Subtraction
less than
less
minus
decreased by
subtracted from
difference between
Multiplication
times
multiplied by
of
as much as
twice
product
Division
quotient
divided by
ratio
Example. Translate each verbal sentence into an equation, using x as the variable.
1. The sum of a number and 6 is 28.
2. The product of a number and 7 is twice the number plus 12.
3. The quotient of a number and 6, added to twice the number, is 7.
Solving an Applied Problem
1. Read the problem, several times if necessary, until you understand what is given and what is to
be found.
2. Assign a variable to represent the unknown value, using diagrams or tables as needed. Write
down what the variable represents. If necessary, express any other unknown values in terms of
the variable.
3. Write an equation using the variable expression(s).
4. Solve the equation.
5. State the answer to the problem. Does it seem reasonable?
6. Check the answer in the words of the original problem.
Example. The length of a rectangle is 5 cm more than its width. The perimeter is five times the width.
What are the dimensions of the rectangle (length and width)? Hint: Recall that P = 2L + 2W .
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Lecture Notes
Chapter 2
Example. A man has $34,000 to invest. He invests some of the money at 5% and the balance at 4%.
His total annual interest income is $1545. Find the amount invested at each rate. Hint: Recall that
I = P rt.
Example. How many pounds of candy worth $8 per lb should be mixed with 100 lb of candy worth $4
per lb to get a mixture that can be sold for $7 per lb?
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Lecture Notes
Chapter 2
Example. How many liters of a 10% alcohol solution must be mixed with 40 L of a 50% solution to get
a 40% solution?
Example. How much water must be added to 20 L of 50% antifreeze solution to reduce it to 40%
antifreeze? (Note: When water is added to a solution, water is 0% of the chemical. Similarly, pure
chemical is 100%)
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Lecture Notes
Chapter 2
Further Applications of Linear Equations
Example. Mohammed has a box of coins containing only dimes and half-dollars. There are 26 coins,
and the total value is $8.60. How many of each denomination of coins does he have?
Example. Two cars leave the same town at the same time. One travels north at 60 mph and the other
south at 45 mph. In how many hours will they be 420 mi apart?
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Example. When Michael drives his car to work, the trip takes
hr. When he rides the bus, it takes
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3
hr. The average speed of the bus is 12 mph less than his speed when driving. Find the distance he
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travels to work.
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Lecture Notes
Chapter 2
Example. Find three consecutive integers such that the sum of the first and twice the second is 122
minus three times the third.
Example. Find the value of x, and determine the measure of each angle. (Hint: the sum of the angle
measures of any triangle is 180◦ ) Draw an acute triangle, top angle (x + 61)◦ , bottom left angle (2x + 7)◦ ,
and bottom right angle x◦
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Lecture Notes
Chapter 2
Linear Inequalities in One Variable
To solve a linear inequality, we use the same properties (meaning, the distributive property, clear the
fractions, and add / subtract terms from side to side) to write equivalent inequalities until the variable
is isolated on one side.
We always REVERSE the original inequality symbol when multiplying and dividing both sides of an
inequality by a NEGATIVE NUMBER.
Example. Solve each inequality. Give the solution set in both interval and graph form.
1. 5x + 3 ≤ 4x − 1
2. −9x < −81
3.
4. 5 < 3x − 4 < 9
5. −3 ≤
9
1
3
(x + 3) + 2 ≤ (x + 8)
4
4
3x + 1
<3
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Lecture Notes
Chapter 2
Example. A rental company charges $5 to rent a leaf blower, plus $1.75 per hour. Marge can spend no
more than $26 to blow leaves from her driveway and pool deck. What is the maximum amount of time
she can use the rented leaf blower?
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Set Operations and Compound Inequalities
Definition. For any two sets A and B, the intersection of A and B, symbolized A ∩ B, is defined as
follows:
A ∩ B = {xx is an element of A and x is an element of B}
Example. Let A = {3, 4, 5, 6} and B = {5, 6, 7}. Find A ∩ B.
Definition. For any two sets A and B, the union of A and B, symbolized A ∪ B, is defined as follows:
A ∪ B = {xx is an element of A or x is an element of B}
Example. Let A = {3, 4, 5, 6} and B = {5, 6, 7}. Find A ∪ B.
Solving a Compound Inequality with ”and”
1. Solve each inequality individually
2. Since the inequalities are joined with and, the solution set of the compound inequality will include
all numbers that satisfy both inequalities in Step 1 (the intersection of the solution sets).
3. If there is NO INTERSECTION (meaning, the two solution sets dont overlap!), then the final
solution set is EMPTY, or ∅.
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Lecture Notes
Chapter 2
Example. Solve the compound inequality and graph the solution set.
1. x + 3 < 1 and x − 4 > −12
2. 2x ≤ 4x + 8 and 3x > −9
3. x + 2 > 3 and 2x + 1 < −3
Solving a Compound Inequality with ”or”
1. Solve each inequality individually
2. Since the inequalities are joined with or, the solution set of the compound inequality includes all
numbers that satisfy either one of the two inequalities in Step 1 (the union of the solution sets).
3. If the two solution sets are DISJOINT, then use the symbol ∪ to write the UNION.
Example. Solve the compound inequality and graph the solution set.
1. x − 1 > 2 or 3x + 5 < 2 + 6
2. 3x − 2 ≤ 13 or x + 5 ≤ 7
3. 3x − 2 ≤ 13 or x + 5 ≥ 7
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Lecture Notes
Chapter 2
Absolute Value Equations and Inequalities
Definition. Absolute Value For any real number a,
because absolute value represents distance, and distance is never negative, the absolute value of a
number is always positive or 0.
Solving Absolute Value Equations and Inequalities
Let k be a positive real number and p and q be real numbers.
1. To solve |ax + b| = k, solve the compound equation
ax + b = k
or
ax + b = −k
The solution set is usually of the form {p, q}, which includes two numbers.
2. To solve |ax + b| > k, solve the compound inequality
ax + b > k
or
ax + b < −k
The solution set is usually of the form (−∞, p) ∪ (q, ∞), which consists of two separate intervals.
3. To solve |ax + b| < k, solve the three-part inequality
−k < ax + b < k
The solution set is of the form (p, q), a single interval.
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Lecture Notes
Chapter 2
Alternative methods for solving inequalities with absolute value(s)
1. Solve the related equation.
2. Separate the number line into regions with the solutions from step 1.
3. For each region, choose a test point and determine whether its value satisfies the original inequality.
4. The solution set includes the regions whose test point value is a solution. If the inequality symbol
is ≥ or ≤, the values from step 1 are included as solutions; if < or >, they are not included as
solutions.
Example. Solve the following.
1. |3x − 4| = 11
2. |3x − 4| ≥ 11
3. |3x − 4| ≤ 11
4. |4x − 1| = |3x + 5|
5. |3x + 2| + 4 = 15
6. |x + 2| − 3 > 2
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Lecture Notes
Special cases
7. |6x + 7| = −5
9. |t − 10| − 2 ≤ −3
8. |x| > −1
10. |x + 2| ≤ 0
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Chapter 2