Solving Inequalities

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An inequality is like an equation, but instead of an equal sign (=) it has one of these signs: < : less than ≤ : less than or equal to > : greater than ≥ : greater than or equal to

What do Inequalities mean? • A mathematical sentence that uses one of the inequality symbols to state the relationship between two quantities.

Graphing Inequalities • When we graph an inequality on a number line we use open and closed circles to represent the number. < < Plot an open circle ≥ ≤ Plot a closed circle

x < 5 means that whatever value x has, it must be less than 5. Try to name ten numbers that are less than 5!

-25 -20 -15 -10 -5 0 5 10 15 20 25 Numbers less than 5 are to the left of 5 on the number line. • If you said 4, 3, 2, 1, 0, -1, -2, -3, etc., you are right. • There are also numbers in between the integers, like 2.5, 1/2, -7.9, etc. • The number 5 would not be a correct answer, though, because 5 is not less than 5.

x ≥ -2 means that whatever value x has, it must be greater than or equal to -2. Try to name ten numbers that are greater than or equal to -2

-2 -25 -20 -15 -10 -5 0 5 10 15 20 25 Numbers greater than -2 are to the right of -2 on the number line. • If you said -1, 0, 1, 2, 3, 4, 5, etc., you are right. • There are also numbers in between the integers, like -1/2, 0.2, 3.1, 5.5, etc. • The number -2 would also be a correct answer, because of the phrase, “or equal to”.

Homework

Solving an Inequality • Solve much like you would an equation. • Always undo addition or subtraction first, then multiplication or division. • Remember whatever is done to one side of the inequality must be done to the other side. The goal is to get the variable by itself.

Properties to Know for Solving Inequalities Addition and Subtraction • Adding c to both sides of an inequality just shifts everything along, and the inequality stays the same. • If a < b, then a + c < b + c Example: Alex has less coins than Billy. • If both Alex and Billy get 3 more coins each, Alex will still have less coins than Billy. • Likewise: • If a < b, then a − c < b − c • If a > b, then a + c > b + c, and • If a > b, then a − c > b − c • So adding (or subtracting) the same value to both a and b will not change the inequality

Properties to Know for Solving Inequalities • Multiplication and Division • When we multiply both a and b by a positive number, the inequality stays the same. • But when we multiply both a and b by a negative number, the inequality swaps over!Notice that a<b becomes b<a after multiplying by (-2)But the inequality stays the same when multiplying by +3 • Here are the rules: • If a < b, and c is positive, then ac < bc • If a < b, and c is negative, then ac > bc (inequality swaps over!) • A "positive" example: Alex's score of 3 is lower than Billy's score of 7. • a < b • If both Alex and Billy manage to double their scores (×2), Alex's score will still be lower than Billy's score. • 2a < 2b • But when multiplying by a negative the opposite happens: • But if the scores become minuses, then Alex loses 3 points and Billy loses 7 points • So Alex has now done better than Billy! • -a > -b • The same is true for division – flip the sign of the inequality if dividing by a negative number

-25 -20 -15 -10 -5 0 5 10 15 20 25 Solve an Inequality w + 5 < 8 - 5 -5 w < 3 All numbers less than 3 are solutions to this problem!

-25 -20 -15 -10 -5 0 5 10 15 20 25 1 step Examples 8 + r ≥ -2 -8 -8 r -10 All numbers greater than-10 (including -10) ≥

-25 -20 -15 -10 -5 0 5 10 15 20 25 1 step Examples 2x > -2 2 2 x > -1 All numbers greater than -1 make this problem true!

-25 -20 -15 -10 -5 0 5 10 15 20 25 2 step Examples 2h + 8 ≤ 24 -8 -8 2h ≤ 16 2 2 h ≤ 8 All numbers less than 8 (including 8)

Be Aware of Cases Involving Multiplying and Dividing Inequalities with Negative Numbers • Multiplication Example • Division Example

One More Case • Solve Inequalities with the variable on both sides

Your Turn…. • Solve the inequality and graph the answer. 1. x + 3 > -4 x > -7 2. 6d > 24 d > 4 3. 2x - 8 < 14 x < 11 4. -2c – 4 < 2 *c ≥-3 noticed in this problem you had to flip the inequality

Summarize:Solving Inequalities

Be sure to know the properties affecting inequalities. • Addition and Subtraction: Adding(or subtracting) c to both sides of an inequality just shifts everything along, and the inequality stays the same. • If a < b, then a + c < b + c • If a < b, then a - c < b - c

Be sure to know the properties affecting inequalities. • Multiplication and Division: When we multiply both a and b by a positive number, the inequality stays the same. • But when we multiply both a and b by a negative number, the inequality swaps over!Notice that a<b becomes b<a after multiplying by (-2)But the inequality stays the same when multiplying by +3 • The same is true for division – flip the sign of the inequality if dividing by a negative number

Let’s look at some Real-Life Application Problems

Real-Life Application Hint: 90 x 6 90% = 540 pts. You are taking a history course in which your grade is based on six 100 point tests. To earn an A in class, you must have a total of at least 90%. You have scored an 83, 89, 95, 98, and 92 on the first five tests. What is the least amount of points you can earn on the sixth test in order to earn an A in the course? 83+89+95+98+92= 457 457 - 457 + T ≤ 540 - 457 T ≤ 83

Example 2 • f/3 ≥ 4 • f/3 ▪ 3 ≥ 4 ▪ 3 • F ≥ 12 To play a board game, there must be at least 4 people on each team. You divide your friends into 3 groups. Write and solve an inequality to represent the number of friends playing the game.

Example 3: • 0.50 x +45 ≤ 50 • 0.50 x +45 -45 ≤ 50 -45 • 0.50 x ≤ 5 • 0.50 x / 0.50 ≤ 5 /0.50 • x ≤ 10 You budget $50 a month for your cell phone plan. You pay $45 for your minutes and 250 text messages. You are charged an extra $0.50 for picture messages. Write and solve an inequality to find the number of picture messages you can send without going over your budget.

Go Forth and Prosper! More Practice available on teacher webpage

Solving Inequalities