Title: EQUATIONS, INEQUALITIES
1EQUATIONS, INEQUALITIES ABSOLUTE VALUE
2CONTENT
- 3.1 Linear Equation
- 3.2 Quadratic Expression and Equations
- 3.3 Inequalities
- 3.4 Absolute value
33.1 Linear Equations
- At the end of this topic, you should be able to
- Define linear equations
- Solve a linear equation
- Solve equations that lead to linear equations
- Solve applied problems involving linear equations
4Equation in one variable
- A statements in which 2 expressions (sides) at
least one containing the variable are equal - It may be TRUE or FALSE depending on the value of
the variable. - The admissible values of the variable (those in
the domain of the variable), if any, that result
in a TRUE statement are called solutions or root. - To solve an equation means to find all the
solutions of the equation
5Equation in one variable, cont
- An equation will have only one solution or more
than one solution or no real solutions or no
solution - Solution set the set of solutions of an
equation, a - Identity An equation that is satisfied for
every value of the variable for which both sides
are defined - Equivalent equations Two or more equations that
have the same solution set.
6Linear Equations
- A Linear Equation in one variable is equivalent
to an equation of the form
- where a and b are real numbers and
- The linear equation has the single solution given
by the formula - Simplify the given equations first, to solve a
linear equations
7Steps for Solving a Linear Equation
- STEP 1 If necessary, clear the equation of
fractions by multiplying both sides by the least
common multiple (LCM) of the denominators of all
the fractions. - STEP 2 Remove all parentheses and simplify
- STEP 3 Collect all terms containing the variable
on one side and all remaining terms on the other
side. - STEP 4 Check your solution (s)
8Solve a Linear Equation
- Solve the following equations
9Solve equations that lead to linear equations
- Solve the following equations
10An equation with no solution
- Solve the following equations
11Translating Written/Verbal Information into a
Mathematical Model
12Solve applied problems involving linear equations
- Example 1
- A total of RM18000 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 2
- Amy grossed RM435 one week by working 52 hours.
Her employer pays time-and-half for all hours
worked in excess of 40 hours. With this
information, can you determine Amys regular
hourly wage?
133.2 Quadratic Expression Equations
- At the end of this topic you should be able to
- Define quadratic expressions and equations
- Solve quadratic equations by factorization,
square root method, completing the squares and
quadratic formula - Recognize the types of roots of a quadratic
equation based on the value of discriminant - Solve applied problems involving quadratic
equations
14Quadratic Equations
- A Quadratic Equation in is an equation equivalent
to one of the form - where a, b and c are real numbers and
- A Quadratic Equation in the form
- is said to be in standard form
-
- 4 ways to solve quadratic equations
- a. Factoring b. Square
root method - c. Completing the square c. Quadratic Formula
15Solve a Quadratic Equation by Factoring
- Solve the following equations
- Repeated Solution / root of multiplicity 2 /
double root - When the left side, factors into 2 linear
equations with same solution
16Solve a Quadratic Equation by the Square Root
Method
- Solve the following equations
17Solve a Quadratic Equation by the method of
Completing the Square
- Adjust the left side of a quadratic equation, so
that it becomes a perfect square (the square of
first degree polynomial). - STEP
18Solve a Quadratic Equation by the Method of
Completing the Square
- Solve the following equations by using completing
the square method
19Solve a Quadratic Equation by the Quadratic
Formula
- Use the method of completing the square to obtain
a general formula for solving the quadratic
equation - Solve the following equations
20Discriminant of a Quadratic Equation
- For a Quadratic Equation
- If
- there are two unequal real solutions
- If
- there is a repeated solution, a root of
multiplicity 2 - If
- there is no real solution (complex roots)
21Examples
- Find a real solutions, if any, of the following
equations
22Application of Quadratic Equations
- Example 1
- The quadratic function
- models the percentage of the U.S. population
f (x), that was foreign-born x years after 1930.
According to this model, in which year will 15
of the U.S. population be foreign-born? -
- Example 2
- The height of projectile at any given time is
given by the equation
, where h is the elapsed time in seconds, and
v is the initial velocity in feet per second. The
constant k represents the initial height of the
object above ground level, as when a person
releases an object 5 ft above ground in a
throwing motion. If the person were on a hill 60
ft, k would be 65 ft. A person standing on a hill
60 ft high, throws a ball upward with initial
velocity of 102 ft/sec. How many seconds until
the ball hits the ground at the bottom of the
hill.
233.3 Inequalities
- At the end of this topic you should be able to
- Relate the properties of inequalities
- Define and Solve linear inequalities
- Define Solve quadratic inequalities
- Understand and solve rational inequalities
involving linear and quadratic expression
24Properties of Inequalities
- If a lt b and b lt c then a lt c
- If a lt b and c is any number, then a c lt b c
- If a lt b and c is any number, then a c lt b c
- If a gt 0 and b gt 0 then a b gt 0
- If a gt 0 and b gt 0 then ab gt 0
- If a lt b then b a gt 0
- If a gt b and a lt b
- If a lt b and a gt b
- If a lt b and c gt 0 then ac lt bc
- If a lt b and c lt 0 then ac gt bc
-
-
reciprocal property - reciprocal
property
25Solving Linear Inequalities
- Solve the following inequality and graph the
solution set
26Solves problems involving linear inequalities
- At least, minimum of, no less than
- At most, maximum of, no more than
- Is greater than, more than
- Is less than, smaller than
27Examples
- Sashas grade in her math course is calculated by
the average of four tests. To receive and A for
the course, she needs an average at least 89.5.
If her current test scores are 84, 92, and 94,
what range of scores can she make on the last
test to receive an A for the course? - A painter charges RM80 plus RM1.50 per square
foot. If a family is willing to spend no more
than RM500, then what is the range of square
footage they can afford?
28Solving Quadratic inequalities
- Step 1 - solve the related quadratic equation
- Step 2 plot the solution on a number line
- Step 3 Choose a test number from each interval
substitute the number into the inequality - If the test number makes the inequality true
- All numbers in that interval will solve the
inequality - If the test number makes the inequality false
- No numbers in that interval will solve the
inequality - Step 4 State the solution set of the inequality
( It is a union of all intervals that solves the
inequality) - If the inequality symbols are or ,
then the values from Step 2 are included. - If the symbols are gt or lt, they are not solutions
29Examples
- Solve the following inequality and graph the
solution set
30Solving rational inequality
- STEP 1 Solve the related equation
- STEP 2 Find all values that make any denominator
equal to 0 - STEP 3 Plot the number found in Step 1 and 2 on
a number line - STEP 4 Choose a test number from each interval
and determine whether it solves the inequality. - STEP 5 The solution set is the union of all
regions whose test number solves the inequality.
If the inequality symbol is or , includes
the values found in step 1 - STEP 6 The solution set never includes the
values found in Step 2 because they make the
denominator equal to 0
31Examples
- Solve the following inequality and graph the
solution set
323.4 Absolute Value
- Equations involving absolute value
- Inequalities involving absolute value
33Solve equations involving absolute value
- Solve the following equation
34Solve inequalities involving absolute value
- Solve the following inequalities. Graph the
solution set
35Application of Absolute value
- The inequality
-
- describes the percentage of children in the
population who think that being grounded is a bad
thing about being kid. Solve the inequality and
interpret the solution
36Thank you