Algebra II For Dummies. Sterling Mary Jane

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how much would she save? I created a spreadsheet and used the formula for an amortized loan (mortgage). I made different columns showing the principal balance that remained (solved for P) and the amount of the payment going toward interest (solved for the difference), and I extended the spreadsheet down for the number of months of the loan. We put the different payment amounts into the original formula to see how they changed the total number of payments and the total amount paid. She was amazed. I was even amazed! She’s paying off her mortgage much sooner than expected!

      

When you rewrite a formula aimed at solving for a particular unknown, you can put the formula into a graphing calculator or spreadsheet to do some investigating into how changes in the individual values change the variable that you solve for (see a spreadsheet example of this in the “Paying off your mortgage with algebra” sidebar).

      Linear Inequalities: Algebraic Relationship Therapy

      Equations – statements with equal signs – are one type of relationship or comparison between things; they say that terms, expressions, or other entities are exactly the same. An inequality is a bit less precise. Algebraic inequalities show relationships between two numbers, a number and an expression, or between two expressions. In other words, you use inequalities for comparisons.

      Inequalities in algebra are less than (<), greater than (>), less than or equal to (< ), and greater than or equal to (> ). A linear equation has only one solution, but a linear inequality has an infinite number of solutions. When you write

, for example, you can replace x with 6, 5, 4, –3, –100, and so on, including all the fractions that fall between the integers that work in the inequality.

      

Here are the rules for operating on inequalities (you can replace the < symbol with any of the inequality symbols, and the rule will still hold):

      ✔ If a < b, then a + c < b + c (adding any number c).

      ✔ If a < b, then a – c < b – c (subtracting any number c).

      ✔ If a < b, then

(multiplying by any positive number c).

      ✔ If a < b, then

(multiplying by any negative number c).

      ✔ If a < b, then

(dividing by any positive number c).

      ✔ If a < b, then

(dividing by any negative number c).

      ✔ If

, then

(reciprocating fractions).

      Notice that the direction of the inequality changes only when multiplying or dividing by a negative number or when reciprocating (flipping) fractions.

      

You must not multiply or divide each side of an inequality by zero. If you do so, you create an incorrect statement. Multiplying each side of 3 < 4 by 0, you get 0 < 0, which is clearly a false statement. You can’t divide each side by 0, because you can never divide anything by 0 – no such number with 0 in the denominator exists.

Solving linear inequalities

      To solve a basic linear inequality, you first move all the variable terms to one side of the inequality and the numbers to the other. After you simplify the inequality down to a variable and a number, you can find out what values of the variable will make the inequality into a true statement. For example, to solve 3x + 4 > 11 – 4x, you add 4x to each side and subtract 4 from each side. The inequality sign stays the same because no multiplication or division by negative numbers is involved. Now you have 7x > 7. Dividing each side by 7 also leaves the sense (direction of the inequality) untouched because 7 is a positive number. Your final solution is x > 1. The answer says that any number larger than one can replace the x’s in the original inequality and make the inequality into a true statement.

      The rules for solving linear equations (see the section “Linear Equations: Handling the First Degree”) also work with inequalities – somewhat. Everything goes smoothly until you try to multiply or divide each side of an inequality by a negative number.

      

When you multiply or divide each side of an inequality by a negative number, you have to reverse the sense (change < to >, or vice versa) to keep the inequality true.

      The inequality 4(x – 3) – 2> 3(2x + 1) + 7, for example, has grouping symbols that you have to deal with. Distribute the 4 and 3 through their respective multipliers to make the inequality into 4x – 12 – 2> 6x + 3 + 7. Simplify the terms on each side to get 4x – 14 > 6x + 10. Now you put your inequality skills to work. Subtract 6x from each side and add 14 to each side; the inequality becomes –2x> 24. When you divide each side by –2, you have to reverse the sense; you get the answer x< – 12. Only numbers smaller than –12 or exactly equal to –12 work in the original inequality.

      

When solving the previous example, you have two choices when you get to the step 4x – 14> 6x + 10, based on the fact that the inequality a < b is equivalent to b > a. If you subtract 6x from both sides, you end up dividing by a negative number. If you move the variables to the right and the numbers to the left, you don’t have to divide by a negative number, but the answer looks a bit different. If you subtract 4x from each side and subtract 10 from each side, you get –24> 2x. When you divide each side by 2, you don’t change the sense, and you get –12> x. You read the answer as “–12 is greater than or equal to x.” This inequality has the same solutions as x< – 12, but stating the inequality with the number coming first is a bit more awkward.

Introducing interval notation

      You can alleviate the awkwardness of writing answers with inequality notation by using another format called interval notation. You use interval notation extensively in calculus, where you’re constantly looking at different intervals involving the same function. Much of higher mathematics uses interval notation, although I really suspect that book publishers pushed its use because it’s quicker and neater than inequality notation. Interval notation uses parentheses, brackets, commas, and the infinity symbol to bring clarity to the murky inequality waters.

      

And, surprise surprise, the interval-notation system has some rules:

      ✔ You order any numbers used in the notation with the smaller number to the left of the larger number.

      ✔ You indicate “or equal to” by using a bracket.

      ✔ If the solution doesn’t include the end number, you use a parenthesis.

      ✔ When the interval doesn’t end (it goes up to positive infinity or down to negative infinity), use +∞ or –∞, whichever is appropriate,

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