Algebra: Chapter 3 Notes
Notes #17: Sections 3-1, 3-2, and 3-6
Section 3-1: Inequalities and their graphs
A. Identifying Solutions of Inequalities
A solution of an inequality is any number that makes the inequality ______.
For example, the solutions of the inequality x < 3 are all: ______
Identifying Solutions by Mental Math
Is each number a solution of x 7? Meaning, does the value of x make the inequality ______?
1.) 9Solution: ______/ 2.)
Solution: ______/ 3.) -5
Solution: ______
Identifying Solutions by Evaluating
Is each number a solution of 6x - 3 > 10? Meaning, does the value of x make the inequality ______?
4.) 3Solution: ______/ 5.) 4
Solution: ______
B. Graphing and Writing Inequalities in One Variable
How many solutions are there to an inequality like m < -3.5? ______
Rather than list solutions, you can use a graph to indicate all of the solutions of an inequality.
When graphing an inequality on a number line, follow these tips:
· Always arrange the final inequality so the variable is on the ______side
· Label your number so that the number in the solution is in the ______of the graph
· Use a ______for < or > and use a ______for ≤ or ≥
· Graph to the RIGHT when a ______or ______and to the LEFT when a ______or ______
6.) x < 3/ 7.) m -2
8.) c > -2
/ 9.) -4 p
Write an inequality for each graph. Variable choice may vary.
10.)/ 11.)
12)
/ 13)
Section 3-2: Solving Inequalities Using Addition and Subtraction
A. Using Addition and Subtraction to Solve Inequalities
**When solving an inequality, you can add and subtract the same number from______without changing the inequality sign**
Solve the following inequalities, then graph the solutions.
14.) x – 3 < 5./ 15.) m – 6 > -4
16.) y + 5 < -7 / 17.)
18.) 3.8 d + 7 / 19.) 12 x – 5
20.) / 21.)
Example (Application)
22.) In order to receive a B in your literature class, you must earn more than 350 points of reading credits. Last week you earned 120 points. This week you earned 90 points. How many more points must you earn to receive a B?Section 3-6: Solving Absolute Value Equations
A. Solving Absolute Value Equations
What does absolute value mean?
Recall that the absolute value of a number is its ______from zero on a number line. Since absolute value represents ______, it can never be ______.
What does solving an absolute value equation mean?
means to find the places on the number line that are ______away from ______.
Solution: ______
means to find the places on the number line that are ______away from ______.
Solution: ______
What does the graph of an absolute value equation look like?
The graph of is below:
Solving Absolute Value Equations:· Get the | | alone
· Write two equations; one ______and one ______
· Solve for x; expect ______answers
· Check both answers by ______
Solve each equation. Check your solution.
1.)(check) / 2.)
(check) / 3.)
(check)
4.)
(check) / 5.)
(check) / 6.)
(check)
Notes #18: Sections 3-6, 3-3 and 3-4
Section 3.6: Solving Absolute Value Equations
1.)(check) / 2.)
(check) / 3.)
(check)
Section 3-3: Solving Inequalities Using Multiplication and Division
When multiplying and dividing the same number to both sides of an inequality,follow these rules:
· If you multiply or divide by a positive number, leave the inequality sign ______
· If you multiply or divide by a negative number, ______the inequality sign.
Explore why:
Solve and graph the solution:
1.)/ 2.)
/ 3.)
4.)
/ 5.)
/ 6.)
7.)
/ 8.)
/ 9.)
10.) / 11.)
/ 12.)
C. Application
13.) Your family budgets $160 to spend on fuel for a trip. How many times can they fill the car’s gas tank if it costs $25 each time?
Section 3-4: Solving Multi-Step Inequalities
D. Solving inequalities with variables on one side
Sometimes you need to perform two or more steps to solve an inequality. Your goal is still the same: to ______the variable on the ______side of the inequality sign.
Solve and graph your solution.
14.) 5 + 4b < 21 / 15.) 2 – 8x > -616.) 8z – 6 < 3z + 12 / 17.) 6z – 15 < 4z + 11
18.) 3x + 4(6 – x) < 2 / 19.) 5(-3 + d) 3(3d – 2)
20.) / 21.)
Notes #19: Section 3-5
Section 3-5: Compound Inequalities
Two inequalities that are joined by the word ______or the word ______form a
______.
A. Solving Compound Inequalities Containing AND
What does it mean?
A solution of an “and” compound inequality is any number that makes ______inequalities true.
Example: Find a solution for the following inequality x < 9 and x > 7 ______
How do I write it?
You can write an “AND” compound inequality as a sANDwich
x -5 and x 7 is the same as ______.
How do I say it?
There are two correct ways to say this:
1) x is ______-5 and ______to 7.
2) x is ______-5 and 7 ______.
How do I graph it?
The solution of this inequality can be expressed with the following graph:
Write a compound inequality that represents each situation. Graph the solution.
1.) All real numbers that are at least -2 and at most 4. / 2.) All real numbers greater than -2 but less than 9. / 3.) The books were priced between $3.50 and $6.00, inclusive.Solve the inequality. Graph the solution.
· When solving “sandwich” problems, it is like you now have three sides of the equation
4.) Solve -4 < r - 5 -1 / 5.) -6 3x < 15 / 6.) -3 < 2x – 1 13B. Solving Compound Inequalities joined by an OR
What does it mean?
A solution of an “or” compound inequality is any number that makes ______inequality true.
Example: Find a few solutions for the following inequality x > 3 or x < -2 _____, _____, _____
How do I write it?
You cannot write an “or” compound inequality as one equation. You must write the solution as _____ inequalities separated by an _____.
How do I say it?
There is only one correct way to say this:
x is ______3 or ______-2.
How do I graph it?
The solution of this inequality can be expressed with the following graph:
Write a compound inequality that represents each situation. Graph the solution.
7.) All real numbers that are less than 0 or greater than 3./ 8.) Discounted tickets are available to children under 7 years old or to adults 65 and older.
9.) Solve and graph the compound inequality
3x+ 2 < -7 or -4x + 5 < 1
/ 10.) Solve and graph the compound inequality
4v + 3 < -1 or -2v + 7 < 1
C. Application
11.) Your test grades in science so far are 83 and 87. What possible grades can you make on your next test to have an average between 85 and 90, inclusive?Notes#20: Section 3-6
Section 3-6: Absolute Value Equations and Inequalities
A. Solving Absolute Value Inequalities
What do absolute value inequalities mean?
means “What numbers are ______2 units away from zero?” Graph the solution:
Is this an “AND” or an “OR” graph?
means “What numbers are ______2 units away from zero?” Graph the solution:
Is this an “AND” or an “OR” graph?
Solving Absolute Value Inequalities:· Get | | alone
· Write 2 equations, one ______and one ______(SWITCH THE SIGN!)
· If use ______
If , use ______
· Graph and solve for x (sometimes, put back in sandwich)
Solve the absolute value inequalities. Graph your solutions.
1.) / 2.)3.)
/ 4.)
5.)
/ 6.)
7.)
/ 8.)
9.) / 10.)
Concept Review:
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