1.4 Sub Chapter Notes

 

Concepts of Inequality < is the symbol meaning that the number on the left is less than the number on the right. > is the symbol meaning that the number on the left is greater than the number on the right. ≤ means that the number on the left is either less than or equal to the number on the right. ≥ means that the number on the left is either equal to or greater than the number on the right. The transitive property means that if A is larger than B and B is larger than C, then A is larger than C. The property also means if A is smaller than B and B is smaller than C, then A is smaller than C. • Adding the same number to both sides of an inequality maintains the inequality. Subtracting the same number from both sides also maintains the inequality. • Multiplying or dividing both sides of an inequality by a positive number maintains the inequality. Multiplying or dividing both sides by a negative number reverses the inequality. • • • • •

In reading inequality symbols, the symbol always opens toward the larger number. Any number that is smaller than one number which, in turn, is smaller than a third number will always turn out also to be smaller than the third number. Likewise, any number that is greater than a number which, in turn, is greater than a third number will also be greater than that third number. You can add or subtract the same amount from both sides of the inequality without changing the nature of the inequality.

Multiplying or dividing by any positive number maintains the inequality that existed before the multiplication. In this example, you start with –3 is less than 2. When you multiply by 5, you get –15 is less than 10 and the inequality is maintained. Multiplying or dividing by any negative number reverses the order of inequality that existed before the multiplication. In this example, you can see that multiplying by –5 alters the inequality.

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One of the most common mistakes in algebra is misapplying inequalities when multiplying or dividing by a negative number.

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Inequalities and Interval Notation <: The number on the left of this sign must be less than the number on the right side. ≤: The number on the left must be less than or equal to the number on the right side. >: The number on the left must be larger than the number on the right side. ≥: The number on the left must be larger than or equal to the number on the right. Interval: The range of values on the number line that are possible solutions for x in a particular problem. ( or o: On a number line, either of these indicates a boundary which x cannot equal. [ or •: On a number line, either of these indicates a boundary that x can equal. Interval Notation: Two numbers enclosed by a parentheses, brackets, or one of each, indicating the range of possible solutions for x. • Infinity: The symbols ∞ and -∞, called positive and negative infinity, are not numbers. They are just symbols that indicate that the number line never ends in either the positive or negative direction. • • • • • • • •

An interval designates the range from which values of x can come. In this case, any number down to, but not including, -3 and up to and including +2 will work as a value for x. The number on the left is the lower boundary of the interval and the number on the right is the upper boundary of the interval. ( or o on a number line indicates that the x can approach that value, but not equal it. In this example, x can approach but not equal -3. [ or • on a number line indicates that the x can both approach and equal that value. Here, x can both approach and equal +2.

Infinity (∞) is never included as an endpoint. There is no way for any variable to actually equal infinity. The interval (-∞,∞) indicates that x can be any and every value in the number system.

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