chapter 2 equations and inequalities

Chapter 2Equations and Inequalities

2.1 More on Solving Equations

When an equation contains fractions or decimals, we can use the multiplication property to eliminate them. The process is called clearing the equation of fractions or decimals.

Objective: Solve equations containing fractions or decimals.

Solving an Equation with Decimals

Solve the equation by clearing it of all decimals first.

Solve the equation by clearing it of all fractions or decimals first.

A

B

C

Objective: Use the principle of zero products to solve equations.

Solving an Equation with Zero Products

( 3)( 2) 0Solve x x 2 ( 1) 0Solve x x

Solving an Equation with Zero Products

( 2 5)(5 1) 0Solve x x

Solving an Equation with Zero Products

D (2 4)( 4) 0x x

Solve the equation using the zero products property.

E (2 16) 0x x

Solving an Equation

Solve the equation

( 4) ( 4)x x x x

Solving an Equation

Solve the equation

3 1( 4) 4(2 5) 4( 4) (2 5)

5 2x x x x

2.2 Using Equations

Objective: Solve problems by translating to equations.

Objective: Solve problems by translating to equations.

A 23-ft cable is cut into two pieces, one three times as long as the other. How long are the pieces?

Herman is doing an experiment that calls for a 40% solution of copper sulfate. He has a 60ml of a solution that is 25% copper sulfate. How many milliliters of a 70% solution should Herman add to obtain the required 40% solution?

Two cars leave the 7-11 at the same time going in the same direction. One car is going 45 mph and the other car is going 58 mph. In 2.5 hours how far apart are they?

A popular music CD was discounted 25% to $8.42. What was the original price?

Write an equation and use it to solve the following

F A tank at a marine exhibit contains 2000 gallons of sea water. The sea water is 7.5% salt. How many gallons, to the nearest gallon, of fresh water must be added to the tank so that the mixture contains only 7% salt?

HW #2.1-2Pg 64 15-41 Odd, 42-46

Pg 70 19-27 Odd

Pg 64 41 Pg 64 44 Pg 70 25 Pg 70 27

Pg 64 39 Pg 64 42 Pg 70 21 Pg 70 27

HW Quiz #2.1-2Thursday, April 20, 2023

2.3 Solving Formulas

Objective: Solve a formula for a specified letter.

A formula is a rule for doing a specific calculation.

Solving a Formula for a specific variable

Solving a Formula for a specific variable

Solving a Formula for a specific variable

Solving a Formula for a specific variable

1 1 1 1Solvefor :x

z y x w

A: Solvefor : 3 5r Q r p

B: Solvefor : rp A p p t

2.4 Solving Inequalities

Objective: Determine if a number is a solution of an inequality and graph the solution set.

If a number occurs to the left of another on the number line, the first number is less than the second, and the second is greater than the first.

2 3 3 2

Objective: Solve and graph inequalities using the addition property.

Objective: Solve and graph inequalities using the multiplication property.

Solving an Inequality

( 3)( 2) 0x x ( 4)( 5) 0x x

Solving an Inequality

30

3xx

40

3xx

G ( 4)( 5) 0x x 4F 0

3xx

HW #2.3-4Pg 72 17-23 Odd, 25-31Pg 77 19-27 Odd, 28-38

Pg 72 26 Pg 72 28 Pg 77 30 Pg 77 34

Pg 72 28 Pg 72 30 Pg 77 32 Pg 77 36

HW Quiz #2.3-4Thursday, April 20, 2023

2.5 Using Inequalities

Suppose that a machinist is manufacturing boxes of various sizes with rectangular bases. The length of a base must exceed the width by at least 3 cm, but the base perimeter cannot exceed 24 cm. What widths are possible?

2.6 Compound Inequalities

A compound inequality is two simple inequalities joined by “and” or “or.”

Objective: Solve compound inequalities.

Statements involving the word “and” are called

conjunctions.For a conjunction to be true

all individual statements must be true

Statements involving the word “or” are called

disjunctions.A disjunction is true when at least one of the statements is

true

Conjunctions2 3x and x

2 3x and x

In mathematics, conjunction is called intersection and is symbolized by the symbol:

Disjunctions

2 3x or x

2 3x or x

In mathematics, disjunction is called union and is symbolized by the symbol:

Solve the following inequalities. Graph your solution on a number line and write them in interval notation.

A 4 3( 2) 2x 3 3

B 1 0 1 52 2

x or x

C 22 50 20 68x D 1 5 1 3x or x

Write and solve a compound inequality that is the conjunction of two disjunctions.

Write an inequality that has no solution and show it has no solution.

Write an inequality whose solutions are all real numbers and show why the solutions are all real numbers.

HW #2.5-6Pg 81 10-18

Pg 85-86 1-31 Odd, 33-45

Solve Write answer in interval notation

-2 < 3x – 4 < 8

Solve Write answer in interval notation

-2 < 3x – 4 or 3x – 4 ≤ 8

Solve Write answer in interval notation

-4 < 3x – 4 < 8

Solve Write answer in interval notation

-4 < 3x – 4 or 3x – 4 ≤ 8

HW Quiz #2.5-6Thursday, April 20, 2023

Row 1, 3, 5

Row 2, 4

2.7 Absolute Value

|2| = |-2| = 2

Objective: Simplify absolute value expressions.

Objective: Find the distance between two points using absolute value.

Objective: Solve and graph equations and inequalities involving absolute value.

Objective: Solve and graph equations and inequalities involving absolute value.

Objective: Solve and graph equations and inequalities involving absolute value.

Objective: Solve and graph equations and inequalities involving absolute value.

Objective: Solve and graph equations and inequalities involving absolute value.

Objective: Solve and graph equations and inequalities involving absolute value.

Objective: Solve and graph equations and inequalities involving absolute value.

| 1| | 2 |x x | 1| | 2 |x x

5 2 1x x

Objective: Solve and graph equations and inequalities involving absolute value.

HW #2.7Pg 91 1-37 Odd, 38-51

Solve

|2x – 3| > 5

Solve

|2x – 3| < 11

Solve

|3x – 2| > 6

|3x – 2| < 10

HW Quiz #2.7Thursday, April 20, 2023

Row 1, 3, 5

Row 2, 4

2-8 Proofs in Solving Equations

Test 1-2

• Part 1-5Questions

• 2-proofs– Any Proof in the HW– Any Proof I did in class

• 1-Closure

• 2 questions with no numbers

Objective: Prove conditional statements.

Conditional Statements

P QAntecedentHypothesis Consequence

Conclusion

To prove a conditional statement you assume the Hypothesis and Show the conclusion must also be true.

Use a two column proof or narrative proof

P QObjective: Write and prove the converse of a conditional statement.

Q PConditional Statement

Converse of the Conditional

Statement

To prove a the converse of a conditional statement you assume the Conclusion and Show the Hypothesis must also be true.

Objective: Solve equations and inequalities by proving a statement and its converse.

P QandP Q Q P

Biconditional

HW #2.8Pg 96-97 1-21 Odd, 23-32

Test Review Topics

• Simplify – Order of Operations• Solve Equations• Exponents – Multiply/Divide – Negative Exponents• Word Problem • Compound Inequalities – Interval Notation• Absolute Value• Sign Charts• Solve for a Variable• Factoring – to solve quadratics

5 2 1| | | |x x

2 2 10 8| |x

20

3 1( )( )

x

x x

x yp

xy

2 7 36 2 x x x

A suitcase of money contains b hundred dollar bills. After the bills are distributed evenly among g federal agents, 8 hundred dollars bills are left over. In terms of b and g, how many hundred dollar bills did each agent get?

If a Citation Jet travels at an average speed of x miles per hour, how many hours would it take the Citation to travel 800 miles?

HW R1-2

• Study Hard

Find the area of an equilateral triangle