Chapter 1 Equations and Inequalities
Chapter 1 Equations and Inequalities
1. Linear Equations
Definition 1 An equation in one variable is a statement in which two expressions, at least one containing the variable, are equal.
Example 1 Solve the equation[pic]
Definition 2 A linear equation in one variable is an equation that can be written in the form
[pic],
where a and b are real numbers and a ≠ 0.
Remark A linear equation is also called a first-degree equation.
Solving Linear Equations in One Variable
1) Clear fractions: Multiply on both sides by the LCD to clear the equations of fractions if they occur.
2) Remove parentheses: Use the distributive property to remove parentheses if they occur.
3) Simplify: Simplify each side of the equation by combining like terms.
4) Isolate the variable: Get all variable terms on one side and all numbers on the other side by using the addition property of equality.
5) Solve: Get the variable alone by using the multiplication property of equality.
Example 2 Solve the equation 3(2x – 4) = 7 – (x + 5).
Example 3 Solve the equation [pic]
Example 4 Solve the equation [pic]
Example 5 Solve the equation [pic]
Example 6 Solve the equation[pic].
Solving for a Specified Variable
All other letters are considered as numbers.
Example 7 Solve PV = nRT for T.
Example 8 Solve [pic] for F.
Steps for Setting Up Applied Problems
1) Read the problem carefully and identify what you are looking for.
2) Assign a letter (variable) to represent what you are looking for, and, if necessary, express any remaining unknown quantities in terms of this variable.
3) Make a list of all the known facts, and translate them into mathematical expressions. Set up the equation.
4) Solve the equation for the variable, and then answer the question, usually using a complete sentence.
5) Check the answer with the facts in the problem.
Example 9 A total of $18,000 is invested, some in stocks and some in bonds, if the amount invested in bonds is half that invested in stocks, how much is invested in each category?
Example 10 Shannon grossed $435 one week by working 52 hours. Her employer pays time-and-a-half for all hours worked in excess of 40 hours. With this information, can you determine Shannon’s regular hourly wage?
2. Quadratic Equations
Definition 1 A quadratic equation is an equation which can be written in the form
[pic]
where a, b and c are real numbers and a ≠ 0.
Remark A quadratic equation written in the form of [pic]is said to be in standard form.
Remark A quadratic equation is also called a second-degree equation.
Solving Quadratic Equations
1) Solving a Quadratic Equation by Factoring
Zero-product Property If [pic], then a=0 or b=0.
Example 1 Solve [pic]
Example 2 Solve [pic]
Remark When the left side factors into two linear equations with the same solution, the quadratic equation is said to have a repeated solution. We also call this solution a root of multiplicity 2, or a double root.
Example 3 Solve [pic]
2) The Square Root Method
If [pic] and [pic], then [pic] or [pic].
Example 4 Solve (a) [pic] (b)[pic]
3) Complete square
Procedure for completing a square
Start Add Result
[pic] [pic] [pic]
Example 5 Determine the number that must be added to each expression to complete the square. Then factor the resulted expression.
(1) [pic] (2) [pic]
Example 6 Solve by completing the square.
(1) [pic]
(2) [pic]
4) The Quadratic Formula
For [pic], we have the quadratic formula
[pic].
[pic] is called the discriminant of the quadratic equation.
1) If [pic], there are two unequal real solutions.
2) If [pic], there is a repeated solution, a root of multiplicity 2.
3) If [pic], there is no solution.
Example 7 Solve [pic]
Example 8 Solve [pic]
Example 9 Solve [pic]
Example 10 Solve [pic]
1.4 Radical Equations; Equations Quadratic in Form; Factorable Equations
Definition 1 When the variable in an equation occurs in a radical (square root, cube root, and so on), the equation is called a radical equation.
To solve a radical equation,
1) Isolate the most complicated radical on one side of the equation.
2) Eliminate it by raising each side to a power equal to the index of the radical.
3) We need to check all answers when working with radical equations.
Example 1 Solve [pic]
Example 2 Solve [pic]
Example 3 Solve [pic]
Definition 2 If an appropriate substitution u transforms an equation in the form
[pic] u ≠ 0
then the original equation is called an equation quadratic in form.
Example 4 Solve [pic]
Example 5 Solve [pic]
Example 6 Solve [pic]
Factorable Equations
Example 7 Solve [pic]
Example 8 Solve [pic]
1.5 Solving Inequalities
Intervals
|Interval |Inequality |Graph |
|Open interval [pic] | | |
|Closed interval [pic] | | |
|Half-open interval [pic] | | |
|Half-open interval [pic] | | |
|Interval [pic] | | |
|Interval [pic] | | |
|Interval [pic] | | |
|Interval [pic] | | |
|Interval [pic] | | |
Example 1 Write each inequality using interval notation.
(a) [pic], (b) [pic], (c) [pic], (d) [pic]
Example 2 Write each interval as an inequality involving x
a) [1,4), (b) (2,[pic]), (c) [2,3], (d) (-[pic])
Properties of Inequalities
• Nonnegative Property: [pic].
• Addition Property for Inequalities:
If a < b, then a + c < b + c.
If a > b, then a + c > b + c.[pic]
• Multiplication Properties for Inequalities:
If a < b and if c > 0, then ac < bc.
If a > b and if c > 0, then ac > bc.
If a < b and if c < 0, then ac > bc.
If a > b and if c < 0, then ac < bc.
• Reciprocal Properties:
If [pic], then [pic].
If [pic], then [pic].
Example 3 (a) If [pic], then
b) If [pic], then
Example 4 (a) If [pic], then
(b) If [pic], then
c) If [pic], then
Solving Inequalities
Example 5 Solve the inequality[pic].
Example 6 Solve the inequality[pic].
Solving Combined Inequalities
1) Keep the variable in the middle.
2) Work with all three expressions at the same time.
Example 7 Solve the inequality[pic].
Example 8 Solve the inequality[pic].
Example 9 Solve the inequality[pic].
Example 10 If [pic], find a and b so that [pic].
1.6 Equations and Inequalities Involving Absolute Value
Equations Involving Absolute Value
If a is a positive real number and if u is an algebraic expression, then
|u| = a is equivalent to u = a, or u = -a.
Example 1 Solve the equation[pic].
Example 2 Solve the equation[pic]
Example 3 Solve the equation [pic].
Inequalities Involving Absolute Value (I)
If a is a positive real number and if u is an algebraic expression, then
|u| < a is equivalent to -a a is equivalent to u < -a or u > a
|u| ≥ a is equivalent to u [pic]-a or u ≥ a
Example 6 Solve the inequality[pic].
Example 7 Solve the inequality[pic].
1.7 Applications: Interest, Mixture, Uniform Motion, Constant Rate Jobs
1) Solve Interest Problems
Simple Interest Formula If a principal of P dollars is borrowed for a period of [pic] years at a per annum interest rate r, expressed as a decimal, the interest [pic]charged is
I = Prt.
Example 1 Candy has $70,000 to invest and requires an overall rate of return of 9%. She can invest in a safe, government-insured certificate of deposit, but it only pays 8%. To obtain 9%, she agrees to invest some of her money in noninsured corporate bonds paying 12%. How much should be placed in each investment to achieve her goal?
2) Solve Mixture Problems
Example 2 The manager of a Starbucks store decides to experiment with a new blend of coffee. She will mix some B grade coffee that sells for $5 per pound with some A grade coffee that sells for $10 per pound to get 100 pounds of the new blend. The selling price of the new blend is to be $7 per pound, and there is no difference in revenue from selling the new blend versus selling the other types. How many pounds of the B grade and A grade coffees are required?
3) Solve Uniform Motion Problems
Uniform Motion Formula If an object moves at an average velocity v, the distance s covered in time t is given by
S = vt.
That is, Distance = Velocity ∙ Time
Example 3 Tanya, who is a long-distance runner, runs at an average velocity of 8 miles per hour (mi/hr). Two hours after Tanya leaves your house, you leave in your Honda and follow the same route. If your average velocity is 40mi/hr, how long will it be before you catch up to Tanya? How far will each of you be from your home?
4) Solve Constant Rate Job Problems
If a job can be done in t units of time, [pic] of the job is done in 1 unit of time.
Example 4
One computer can do a job twice as fast as another. Working together, both computers can do the job in 2 hours. How long would it take each computer, working alone, to do the job?
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