§2.1 – LINEAR EQUATIONS in One VARIABLE
When a variable in an equation is replaced
by a number that gives both sides of the equation
the same equal value, that number is called a solution of that equation. To
solve an equation
is to find the solution set of that equation; the solution set is denoted by { …
}.
| Ex. 1: Solve. |
Ex. 2: Solve. |
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When solving equations, if the variable terms on both
sides completely “disappear” and:
(case 1) a FALSE statement is obtained, such as “5 = 6” or “0 = 8”, then the
original has NO
SOLUTION. State so, or use Ø ; or,
(case 2) a TRUE statement is obtained, such as “0 = 0” or “5 = 5”, then the
original has ALL
REAL NUMBERS as solutions. State so, or use R.
| Ex. 3: Solve. |
Ex. 4: Solve. |
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§2.2 – FORMULAS
Formulas are short equations that contain 2 or more variables. (The variables
are to
represent real-life measured quantities; thus, formulas express relationships
among these quantities.)
Some formulas you should be familiar up to now are:
• Simple Interest : 
• Perimeter and Area of a rectangle: 
• Distance: 
, where t is time (in hours) and r is the average speed
• Volume of a cubic solid : 
More formulas will appear along the course of the semester.
Solving for a specified variable in a formula
takes the same procedure as in
solving
equations (see Section 2.1) – the major difference is, we usually can’t combine
terms because
terms in formulas are usually not alike.
Step 1: CLEAR the fraction(s) by multiplying BOTH sides of the equation by the
LCD .
Step 2: Use the DISTRIBUTIVE property to remove parentheses.
Step 3: ISOLATE the term containing the specified variable – everything else
goes to other side.
Step 4: Use the DIVISION property to leave the specified variable alone.
§2.3 – APPLICATIONS of LINEAR EQUATIONS
GENERAL 6- step STRATEGY :
1) READ and re-read the problem – focus on the
question (“what to solve”).
2) ASSIGN A VARIABLE, like x , to represent the unknown value . Express
other unknown
values in terms of that variable. Note: Only 1 variable is used for the
entire problem!
3) TRANSLATE the problem into one equation, using key words, tables or
diagrams.
4) SOLVE the equation.
5) INTERPRET the result – Does your result make sense?
6) WRITE the answer statement in the words of the original problem. |
Ex. 1: In a blueprint of a rectangular room, the length is
2 inches greater than twice the width.
Find the dimensions if the perimeter is 40 inches.
Ex. 2: A man has $34,000 to invest. He invests some of the
money at 5% and the remainder at
4%. His total annual interest income is $1,545. Find the amount invested at each
rate.
Ex. 3: How many pounds of candy worth $8 per lb should be
mixed with 100 lbs of candy
worth $4 per lb to get a mixture that can be sold for $7 per lb?
Watch out! When pure water is added to a chemical
(acid, alcohol, antifreeze, etc.) solution,
water is 0% of the chemical. Similarly, pure chemical solution is 100%
chemical. |
Ex. 4: How much water must be added to 20 liters of 50%
antifreeze solution to reduce it to
40% antifreeze?
§2.4 – FURTHER APPLICATIONS of LINEAR EQUATIONS
Ex. 1: Find three consecutive integers such that the sum of the first and twice
the second is 122
minus three times the third.
Ex. 2: 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 miles apart?
Ex. 3: Mom has a box containing only dimes and quarters.
There are 26 coins, and the total
value is $4.85. How many of each denomination of coin does she have?
§2.5 – LINEAR INEQUALITIES in 1 VARIABLE
Reminder: Inequalities symbols are: < , > , ≤ , ≥ , ≠ .
While linear equation has particular solutions (see Section 2.1), a linear
inequality has many
solutions, thus a solution set to an inequality is always necessary. Let’s first
review the
interval notation of a solution set:
• List the starting value of the interval, then “comma”, then the ending value
of the
interval, in exactly the Left-to-Right order of the values on a number line.
• Use a square bracket , such as [ or ] , when a value is included/contained in
the interval.
• Use a parenthesis, such as ( or ) , when a value is NOT included/contained in
the interval.
Note: The value of the “infinity, ∞ ” is not a definite number, so always use a
parenthesis for
∞ and - ∞ (because, “how can we contain an unknown number?”).
Ex. 1: Given an inequality, graph and write the solution set in interval
notation:
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.
Watch out! We always REVERSE the original
inequality symbol when multiplying and
dividing both sides of an inequality by a NEGATIVE NUMBER. |
Ex. 2: Solve the inequality. Graph the solution set and
write it in interval notation.
Interval Notation: …………………
Ex. 3: Solve the inequality. Graph the solution set and
write it in interval notation.
Interval Notation: …………………
Ex. 4: Solve the inequality. Graph the solution set and
write it in interval notation.
Interval Notation: …………………
Ex. 5: You have scores of 92, 90, and 84 on your first
three tests. What score must you make on
the fourth test to keep an average of at least 90 on the tests?
