We will use a variable to represent the unknown quantity and insert the necessary symbols.

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1 Chapter 3.1 Write and Graph Inequalities When you hear words in a problem that state at least, at most, is less than, is greater than, is less than or equal to, is greater than or equal to or is not equal to you are dealing not with a numeric solution to your problem but instead an inequality. An inequality is a statement that compares two expressions or quantities that may not be equal. It uses one of the comparison symbols: >, <,,,. To write an inequality we are going to carefully read what the problem asks and observe key words associated with the inequality symbols. We will use a variable to represent the unknown quantity and insert the necessary symbols. In our example of the car wash the unknown or variable is the number of cars it will take to reach the fundraising goal. Let n = the number of cars. Each car wash is $12. To determine how much they will make you need to multiply the number of cars that come through the car was by the price they will pay. So we represent the money made by 12n. The school wants to raise at least $1500. That means they can raise $1500 or they can raise more, but they do not want to raise less than that so we use the inequality symbol. Our inequality equation is: 12n You can also write the expression as n. You can reverse the order of your expressions as long as you reverse your inequality symbol. Inequalities such as a -5 have many solutions and it would be impossible to list them all in roster notation. We can graph the inequality {a a -5}. -5 will be our boundary point. Since it is included in our solutions set we will represent the boundary point with a solid filled in point on the number line at -5 and a shaded arrow that extends to the right. If our inequality was a > - 5 we would use an open circle to represent our boundary point since -5 in not part of the solution however every value to the right of -5 in included. When graphing it is easier to always have the variable on the left side of the inequality. To do this you read the inequality backwards. Reverse the order of the numbers and flip the symbol. Always read the inequality from the variable. 3 < n is read from the variable as: n is greater than 3, so the inequality can be rewritten as n > 3. We have represented the solution set to an inequality with set builder notation: {a a -5}. We can also use interval notation. Interval notation shows the endpoints of a solution set. All real numbers between those endpoints are in the interval. {a a -5} set builder notation, [-5, ) interval notation Symbols for interval notation are: [ ] means included or closed ( ) means not included or open means that the interval continues endlessly in the positive direction - means that the interval continues endlessly in the negative direction We can express the solution to an inequality with words, a graph, set builder notation, and interval notation.

2 Chapter 3.2 Solve Inequalities using Addition and Subtraction When solving inequalities, just like equations you want to first combine like terms. Not only can we solve inequalities we can graph the solution set for the inequality. Once we solve an inequality and we graph it we can check the validity of our solution set. To check an inequality you must choose a value that is in the solution set and check to see if the inequality holds true. Then you must choose a value that would not be included in the solution set and make sure that the value proves to make the inequality false. Make sure when graphing inequalities you check your boundary points before you move on. If you have the symbols <, > you want the boundary point to be an open circle. If your solution contains the symbols, your boundary point will be a filled circle.

3 Chapter 3.3 Solving Inequalities Using Multiplication and Division Division property of inequality which states: If a, b, and c are real numbers, c is positive and a < b then a/c < b/c If a, b, and c are real numbers, c is negative and a < b then a/c > b/c You can rewrite this property in the same way for >,,. The Trick with the division property of inequality is that when you divide by a negative number you have to reverse the inequality symbol. That is because on a number line the smaller absolute value of a negative number is actually the greater number. When we solve an inequality the inequality includes all numbers contained in the solution set; however in some problems only whole number solutions make sense in this inequality. To solve a division inequality we will use the multiplication property of inequality which states: If a, b, and c are real numbers, c is positive and a < b then ac < bc If a, b, and c are real numbers, c is negative and a < b then ac > bc Again you must remember to reverse the inequality symbol when multiplying by a negative number. To check the solution chose a value that is included in the solution set and one that is not. Remember: You only reverse the inequality symbol when you divide or multiply both sides of an inequality with a negative number.

4 Chapter 3.4 Solve Multi Step Inequalities To solve the inequality we have to isolate the term containing the variable just like when we solved equations. Again we must follow our order of operation in reverse. Although many solutions make the inequality true, not all solutions make sense based on what the inequality is asking us to solve for. Let's solve 3(2y+1) The first step is to use the distributive property. Compare this to solving 4(-2n+8) > -16 With the first you use the distributive property initially, with the second you use the division property of inequality first to solve the inequality. When we solve equations or inequalities with fractions we can begin by eliminating the fractions on both sides by multiplying by the LCD. Solving multi-step inequalities is very similar to solving multi step equations. The biggest difference is going to be remembering to switch the sign when you multiply or divided both sides of the inequality by a negative number and remembering that to check your solution you must substitute in a value that would be in the solution set and a value that would not be included in the solution set. Remember that solutions to inequalities can be expressed three different ways: Set builder notation graphing interval notation

5 Chapter 3.5 Solve Compound Inequalities A compound inequality is two simple inequalities joined by the words AND or OR. A compound inequality joined by the word AND is called a conjunction. Solutions that solve both parts of the inequality are solutions to the conjunction. To graph a conjunction: Draw and label a number line that includes the boundary points. The boundary points are the points where you start the graph Graph the first inequality On the same number line graph the second inequality Identify the region where the two graphs overlap. This is the graph of the solution set. The intersection satisfies both inequalities. We can then write the solution set of the conjunction using set builder notation. If a, b and c are real numbers, a < b and b < c then a < b < c. You can solve compound inequalities algebraically and represent the solution set graphically or using symbolic notation. Try example 2 x Read as 2 x + 6 AND x Solve both inequalities, graph the solution set of each and where the two graphs intersect gives you the solution set. To check try a value that is in the solution set to see if it satisfies both conditions of the inequalities. Write the solution as set builder notation and interval notation. Remember that the boundary points in interval notation are represented by [ ] and ( ). [ ] means that the boundary point is included in the solution, ( ) means the boundary point is not included in the solution set. Two Simple inequalities may be joined by the word OR as well. This type of compound inequality is called a disjunction. Solutions that satisfy at least one part of this type of compound inequality are the solution to the disjunction. When we speak of conjunction it is very similar to speaking of the intersection of two sets, when we speak about a disjunction we are speaking about the union of two sets. You can solve compound inequalities that are disjunctions algebraically as well. Use your properties of inequality to solve both inequalities. Your solution set can be written graphically, in set builder notation and in interval notation with the U symbol between the two options.

6 When testing a compound inequality that is a disjunction it is best to test a point from each of the three parts the number line is divided into. You can write a compound inequality from a graph. First identify the boundary points from the graph. Determine if they are included or excluded from the solution set. Use the inequality symbol that corresponds to the direction of the arrow. 3 conditions must be met for an inequality written in the form: {-3 x < 2} x must be -3 x must be < 2-3 must be < 2 In a conjunction all three are in fact true. A disjunction cannot be written this way. If our solution set is x > 2 OR -3 x, {-3 x >2} should prove that: x must be -3 x must be > 2-3 must be > 2 and this condition is not true. You must use the word OR when representing the solution set to a disjunction. To graph a compound inequality joined by the word AND (a conjunction) 1. Draw and label a number line that includes the boundary points. The boundary points are the points where you start the graph. 2. Graph the first inequality 3. On the same number line graph the second inequality. 4. Identify the region where the two graphs overlap. This is the graph of the solution set. The intersection satisfies both inequalities. To graph a compound inequality joined by the word OR (a disjunction) Draw and label a number line that includes the boundary points. Graph the first inequality On the same number line graph the second inequality. Identify the union of the two graphs. This is the graph of the solution set. The union satisfies both inequalities.

7 Chapter 3.6 Solve Absolute Value Inequalities The absolute value of any number is the distance that number is from 0 on the number line. Inequalities that contain an absolute value can be written as compound inequalities. When a > 0 x = a means that x is a units from 0, x = a or x = -a x < a means that a is less than x units from 0, x > -a AND x < a (this is true for ) Example x < 10, x < 10 and x > -10 x > a means that a is more than x units from 0, x > a, OR x < -a (this is true for ) Example x > 10, x >10, x < -10: If you have an absolute value equation written with a < or symbol it is set up as a conjunction If you have an absolute value equation written with a > or symbol it is set up as a disjunction Principle for Solving an Absolute Value Inequality Isolate the absolute value expression Write the statement as a compound inequality Solve the two simple inequalities Graph the solution set and check