Inequalities The Prefix in means not. Incorrect, Inflexible Equations which have solutions are equal to a specific value, or number: 2x = 8 can only equal 4; no other number will satisfy this equation. Inequalities, however, can have many answers. They are not equal to a specific value. When solving inequalities, we are solving for a range of numbers, not just one. Lets look at some examples of inequalities
InequalitiesLook at, and think about, the following signs:The problem is, none of these signs say whattheyre really supposed to say. Not only that,they are all incorrect. To be correct, they neededto include an inequality.
Inequalities Lets put this sign in mathematical terms:Let h = the height required to use the ride. The signsays you must be 46" tall, therefore h = 46"According to the sign, if youre not 46" tall, you cannotride. But how many people are exactly 46" tall? What they really meant to say was... You must be at least 46" tall, or in mathematical terms... Your height must be equal to or greater than 46". This is our inequality. Our solution is not a single number, but a range of numbers.
InequalitiesThis sign obviously refers to the drinking age. Butthe sign states that even 22 year olds, or 75 yearold people cannot enter. The two words missinghere are: at leastIn mathematical terms, the drinking age is: Equal to or greater than 21 d > 21
InequalitiesAs far as the signs are written: Incorrect Correct
Solving InequalitiesThe process of solving Inequalities is the same asequations except for one rule(which well get tolater), and how inequalities are shown graphically. Less Than; shown with an open circle on number line; x < -4 Less Than or equal to; shown with closed circle on number line; x < -4
Solving InequalitiesGreater Than; shown with an open circleon number line; x > -4 Greater Than or equal to; shown with a closed circle on number line; x < -4
Solving InequalitiesBasic Inequalities1. Write the inequality shown below x<3 x>0 -5 < x < 2
InequalitiesGraphing InequalitiesDraw a number line and graph the following:1. 1 <x < 8 2, -2 < x < -1 3. -5 < x < 2
Solving Inequalities Solve for x and Graph 1. 6x - 7 < 5 2. 4(x - 2) > 20 3. x - 8 < - 6 1. x < 2; Graph x>3 x<2 And now the one difference between equations & inequalities: Solve for x and Graph4. -2x < 4; When multiplying or dividing by a negativecoefficient, you must switch the sign4. -2x < 4; -2x/-2 > 4/-2; x > -2
InequalitiesThink about the rule for example 4 with numbers in there,instead of variables. -2 < 4You know that the number four is larger than the numbernegative two: 4 > -2.Multiplying through this inequality by –1, we get –4 < –2,which the number line shows is true: If we hadnt flipped the inequality, we would have ended up with "–4 > –2", which clearly isnt true. When multiplying or dividing a negative coefficient, you must flip the sign for the inequality to remain true.
Solving InequalitiesLast 2 Practice Problems;Solve & Graph on Number Line 5. x - 12 < -6 6. 6 - 2x > - x 5. x - 12 < -6; 5. x < 6; +12 +12 6. 6 - 2x > - x +2x +2x 6 > x; x < 6