Unit 2 Review

Unit 2 Review

Algebra/Geometry I

Unit 2 Review:

• Ordering numbers from least to greatest as well as rounding

• Plotting points on a number line

• Solving inequalities and compound inequalities

• Graphing inequalities on a number line

• Word problems

Multiple Choice

Put these numbers in order from least to greatest:

 $-\frac{8}{3}$  , 2.5,  $\frac{15}{9}$  ,  $-\sqrt{18}$  , -1.3

• -8/3, 15/9, -1.3,  $-\sqrt{18}$ , 2.5

•  $-\sqrt{18}$ , -8/3, -1.3, 15/9, 2.5 

• 2.5, 15/9, -1.3, -8/3,  $-\sqrt{18}$  

• -8/3, -1.3, $-\sqrt{18}$ , 15/9, 2.5 

What is an inequality?

• The word inequality simply means a mathematical expression in which the sides are not equal to each other. Basically, an inequality compares any two values, and show that one value is less than, greater than or equal to the value on the other side of the equation.

• These inequality symbols are: less than (<), greater than (>), less than or equal (≤), greater than or equal (≥) and the not equal symbol (≠).

• Inequalities are used to make a comparison between numbers and to determine the range or ranges of values that satisfy the conditions of a given variable

Positive Operations on Inequalities

• The inequality symbol does not change when the same number is added on both sides of the inequality.

• Subtracting both sides of the inequality by the same number does not change the inequality sign.

• Multiplying both sides of an inequality by a positive number does not change the inequality sign.

• Dividing both sides of an inequality by a positive number does not change the inequality sign.

Negative Operations on Inequalities

• Multiplying both sides of an inequality equation by a negative number changes the direction of the inequality symbol. 

• Similarly, dividing both sides of an inequality equation by a negative number changes the inequality symbol.

Solving Inequalities

• Just like linear equations, inequalities can be solved by applying similar rules and steps with a few exceptions. The only difference when solving linear equations is an operation that involves multiplication or division by a negative number. Multiplying or dividing an inequality by a negative number changes the inequality symbol.

• Linear inequalities can be solved using the following operations: Addition, Subtraction, Multiplication, Division, and Distribution of property

How to Graph a Linear Inequality

• Rearrange the equation so "y" is on the left and everything else on the right.

• Plot the "y=" line (make it a solid line for y≤ or y≥, and a dashed line for y< or y>)

• Shade to the right for a "greater than" (y> or y≥) or shade to the left for a "less than" (y< or y≤).

Multiple Choice

 $x>-5$  

Which inequality matches this graph?

Multiple Choice

Match this inequality to its correct graph:  $-5x<-10$  

Multiple Choice

Match this inequality to its correct graph:  $5x-7>6x-5$  

Multiple Choice

Match this inequality to its correct graph: $12\ge6x$  

Multiple Choice

Match this inequality to its correct graph:  $4\left(x-3\right)\ge-4$  

Compound Inequality

• A compound inequality is a sentence with two inequality statements joined either by the word “or” or by the word “and.”

•  “And” indicates that both statements of the compound sentence are true at the same time. It is the overlap or intersection of the solution sets for the individual statements.

•  “Or” indicates that, as long as either statement is true, the entire compound sentence is true. It is the combination or union of the solution sets for the individual statements. 

• A compound inequality that uses the word “and” is known as a conjunction.

How to solve Compound Inequalities

• "And": Solve each inequality separately. Since the joining word is “and,” this indicates that the overlap or intersection is the desired result. Then solve like a regular inequality.

• "Or": Solve each inequality separately. Since the joining word is “or,” combine the answers; that is, find the union of the solution sets of each inequality sentence. Then solve like a regular inequality.

Multiple Choice

Solve the compound inequality:

 $-3<\frac{1}{3}x-7<10$  

• -2 < x < 37

• -30 < x < 9

• 12 < x < 51

• No Solution

Multiple Choice

Solve the inequality: 5 - 3x > 23 OR 2x - 3 > 23

• x > -6 OR x > 13

• x < -6 OR x > 13

• x < -6 OR x < 13

• x > -6 OR x < 13

Word Problems

• Identify the Problem: Once you identify the problem, you can determine the unit of measurement for the final answer.

• Create an Equation: Translate any of the math terms into math symbols.

• Solve the Problem: Use Order of Operations as well all of the rules we have learned for solving inequalities

• Verify the Answer: Check if your answer makes sense with what you know.

Multiple Choice

Alfred is on the on ramp of the highway. As he enters the highway, he speeds up an additional 25 mph, but stays within the highway's speed limit of 65 mph. The inequality s + 25≤ 65 represents his original speed in miles per hour (mph). Which of the following could have been Alfred's original speed?

• 90

• 25

• 2.6

• 40