1) 2) Algebra (3-2) Solving Inequalities with Additon and Subtraction

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1 Algebra (3-2) Solving Inequalities with Additon and Subtraction N# The Equality Properties of Addition and Subtraction also apply to INEQUALITIES. If you or the same value to each side of an inequality, you create an Inequality. Solve and graph the following: 1) 2) 3)

2 Algebra: (3-3) Solving Inequalities with Multiplication/Division Notes # The Equality Properties of Multiplication and Division also apply to INEQUALITIES. IMPORTANT!! IF YOU MULTIPLY OR DIVIDE BY A NEGATIVE NUMBER TO SOLVE, FLIP THE INEQUALITY SYMBOL TO ITS OPPOSITE. Solve the following problems. Graph and check (the shading). 2) 2) 3) 4) 7) Write an inequality for this situation, and solve it. Answer in a sentence. Your family budgets a maximum of $160 to spend on gas for a road trip. How many times can they fill the car s gas tank if it costs $25 each time?

3 Algebra: (3-3) Solving Inequalities with Multiplication/Division Notes # The Equality Properties of Multiplication and Division also apply to INEQUALITIES. IMPORTANT!! IF YOU MULTIPLY OR DIVIDE BY A NEGATIVE NUMBER TO SOLVE, FLIP THE INEQUALITY SYMBOL TO ITS OPPOSITE. Solve the following problems. Graph and check (the shading). 3) 2) 3) 4)

4 7) Write an inequality for this situation, and solve it. Answer in a sentence. Your family budgets a maximum of $160 to spend on gas for a road trip. How many times can they fill the car s gas tank if it costs $25 each time?

5 Algebra (3-4) Multi Step Inequalities Notes # Remember the steps: 1. Clear decimals and fractions (if it makes sense!) 2. Eliminate parentheses by distributing. 3. Combine like terms on each side of the inequality. 4. Collect variables on one side FIRST, then numbers on the other side. 5. Undo addition or subtraction 6. Undo multiplication or division REMEMBER: FLIP the inequality if you multiply or divide by a negative to solve! (The coefficient is negative.) 7. Graph, check shading, write answer in set notation. Examples: 1) 3x ) 5 2(4m 7) 15

6 3) 6( x 4) 7(2x 3) 4)

7 Algebra (3-5) Part 1: Compound Inequalities AND Notes # Two inequalities that are joined by the word AND or the word OR form what is called a compound inequality. An example of this is This compound inequality can also be written as follows: The inequality is read as x is greater than or equal to 3, and less than or equal to 11. It can also be read x is between 3 and 11, inclusive. A compound inequality containing the word AND is true only if BOTH of the inequalities is true. The variable must satisfy BOTH conditions. For example, 6 is a solution of the above compound inequality. Why? Finally, the graph of a compound inequality containing the word AND is where the graph of the two inequalities overlap. This is called the intersection of the graph, and this compound inequality is called a conjunction. Examples: Write a compound inequality that represents each situation. Graph your solution using the rough draft method. From the graph, write your final solution in set notation. 1. All real numbers x that are at least 1 and at most The books were priced between $4.00 and $8.50, inclusive. 3. Today s temperature will be above 72 degrees, but not as high as 78 degrees.

8 To solve a compound inequality: 1. Split it into two inequalities, joined by the word AND, and solve each. Use a T-table. 2. Graph your solution using the ROUGH DRAFT method above the number line. 3. Put the resulting INTERSECTION on your number line. 4. Write your answer in set notation

9 3-5 Compound Inequalities, Part 2 Notes # Compound Inequality - Two inequalities joined by and or or And inequalities can also be written as a single conjunction: x > - 4 and x < 6 is the same as 4 < x < 6 Or inequalities are called disjunctions, and there is only one way to write them. To Solve and graph: 1. Separate each compound inequality into its TWO PARTS 2. Solve each inequality for the variable. 3. Graph each solution above your number line. (Rough draft it s required!) Remember: < or > is an open dot; or is a solid dot 4. Graph the intersection or union the two inequalities on number line. If joined by and - Graph the intersection of the two graphs. It is where the graphs overlap. It is usually between the two numbers, but not always. If joined by or - Graph the union of the two graphs. Combine the two graphs into one. It usually goes in opposite directions, but not always. 5. Write your solution in SET NOTATION, for example: { x: x > - 4 and x < 6 }, which is read the set of x values such that x is greater than 4 and less than 6.

10 Solve and Graph: 1) 4 r 5 1 2) x < 3 or x > 7 3) 4v + 3 < 5 or 2v + 7 < 1 4) 2x + 7> 3 or 3x 4 5

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12 Algebra (3-6) Absolute Value Equations Notes # Absolute Value- The distance (# of steps) a number is from zero on the number line. Distance is always positive, so is the absolute value of any number. It is NOT the opposite of the number! Examples: 5 is the distance from zero to 5 on the number line: 5 = 5 5 is the distance from zero to 5 on the number line: 5 = 5 Use one of the following to solve open sentences involving absolute value: I. If x c, then x c or x c. x is all numbers that are c units from zero on the number line. II. To solve absolute value equations: 1) Isolate the absolute value FIRST 2) Now Split into 2 separate equations without absolute value symbols as follows: First Equation A = positive OR Second Equation = negative 3) Solve each equation. 4) Check your solutions! NO SOLUTION is a valid answer to these equations. Can you think of why that would be? ) x ) 5 f 45

13 3) 2p ) 4 3 w 2 5) 2 7d 14 5) 3 c 2 5

14 3-6 Absolute Value Inequalities Notes Remember, there are TWO answers to every absolute value equation, because there are two numbers that are c units from zero on the number line. One is negative, the other positive. This is true for absolute value inequalities too! So you will have two inequalities to solve, one less than, one greater than. Fun! Here s the rule: To solve A b, solve A b and A b (same as b A b ) Hint to remember: LESS ThAND To solve A b, solve A b OR A b Hint to remember: GREATOR than Solve and graph. Write the solution last, from your graph. 1) v 3 4 2) w 2 5

15 3) 3 4x 5 6 4) 2x 7 2