Section 2.5 Formulas and Additional Applications from Geometry Section 2.6 Solving Linear Inequalities Section 7.1 Evaluating Roots Section 2.5 Formulas and Additional Applications from Geometry Definition of a Formula: A formula is an equation in which variables are used to describe a relationship. Formulas from Geometry to remember: 1. Perimeter of a Square Perimeter measures the distance around an object 2. Perimeter of a rectangle 3. Area of a square Area measures the surface measured by the figure 4. Area of a rectangle 5. Volume of a rectangular box Solving a formula for one variable, given the values of the other variables. 1. Find the value of the remaining variable in each formula. I = Prt; I = $246, r = 0.06, t = 2
2. P =2l + 2w; P=126, W = 25 Using a formula to solve an applied problem Here we will look at word problems that include geometric figures. It is a good idea to draw a diagram whenever possible. 1. A farmer has 800 m of fencing material to enclose a rectangular field. The width of 2. The perimeter of a rectangle is 36 yd. The width is 18 yd less than twice the length. Find the length and width of the rectangle. Solving equations for a variable Here you will treat variables as if they are constants, hence, allowing you to rewrite an equation solved for any variable in the equation. 1. Solve I =Prt for t
2. A = p + prt for t Section 2.6 Solving Linear Inequalities In this section we will look at Graphing intervals on a number line Solving linear inequalities Using inequalities to solve applied problems We saw the inequalities in a previous section < > In this section, we will discuss how to solve a linear inequality. For inequalities, the solution will not be a single value for the variable that makes the equation true; instead in inequalities, there will be infinitely many values or in other words, an entire interval of values that makes the equation true. For example if x < 5, infinitely many numbers substituted in for x makes the inequality true. For example, -1.3, 0, 4.999, -50, 3.4, are all less than 5, and thus make the inequality true. The solution to an inequality is called a solution set. There are two ways to represent a solution set 1. Graphing on a number line 2. Writing it in interval notation.
We will discuss both. The first way to express the solutions to an inequality is by graphing them on a number line. We will just practice graphing inequalities on a number line before we learn to solve them. Here are some examples. 1. x 3 2. x < 3 3. x > 4 4. 5 < x 3 4 Notes on graphing linear inequalities: 1. Use an open circle to indicate that an endpoint is not included (used with > or < ). 2. Use a closed circle to indicate that an endpoint is included in a solution set (used with or
Interval notation Interval Number line Interval notation x < b x b x > a x a a < x < b a x b a < x b a x < b Linear Inequality in One Variable A linear inequality in one variable can be written in the form Ax + B < C where A, B, and C are real numbers, with A 0 Most of the rules you learned for equalities also hold true for inequalities; You can add or subtract any number to/from both sides You can multiply or divide by any non-zero number. But there is one speciall rule we need to careful of!!! Are you ready??? IF YOU MULTIPLY OR DIVIDE BY A NEGATIVE NUMBER YOU MUST FLIP THE INEQUALITY. Let s look at a numerical example It is true that -2 < 1 if I multiply by a POSITIVE NUMBER, let s say 2: 2( 2) < 2( 1) 4 < 2 Which is still true. However, if we multiply by -2 2 < 1 2 ( 2) < 2(1 ) 4 < 2 Ohhh that isn t true. But if you reverse the inequality Currently it says an untrue statement 4 < 2
If you flip the inequality symbol 4 > 2 it is now true So when you multiply or divide by a negative number you must remember to flip the inequality Solve each inequality and graph the solution set 1. 1 + 8r < 7r + 2 2. 9x < 18 3. 7 x 6 + 1 5x x + 2 4. 2 3x 1 8
Application: Maggie has scores of 90, 96 on her midterms, 86 as her quiz average, and 95 as her homework average. The midterms are each worth 25% of her grade, the quizzes are worth 10% of her grade, the homework is worth 10% of her grade and the final is worth 30% of her grade. What score must she receive on the final to get an average of at least 90%? Section 7.1 Evaluating Roots First let s remember what it means to square a number: 2 2 if a = 6, then a = 6 = 6 6 = 36 If we wish to work backwards, then we would have to take what is called the square root. 2 If a = 36, then a =? Well this actually has two answers, a = 6 or a = -6 To find the square root of a number you must find the number, when multiplied by itself, gives the number under the square root symbol. Each number has two square roots, a positive and a negative. The positive square root is called the principal square root and is written as a, and the negative square root is a
So what does this mean: If I give you 64 there is only one correct answer: 8 If I give you 64 there is only one correct answer: -8 This is a fact that is often overlooked. There is a lot of confusion between this 2 and the process of solving x = a. We haven t discussed this yet, but please pay attention to the fact that if there is no sign in front of the square root, the answer is a positive number; if there is a negative in front of a square root, the answer is a negative number. A few other things to note: 1. 0 = 0 2. Not all numbers have square roots. You cannot take the square root of a negative number (at least not in the real number system). 1. 16 2. 169 3. 36 25 4. 64 Squaring Radical Expressions: Squaring a square root removes the radical.
Square the following: 1. 64 2. 41 3. 2x 2 + 3 Higher Roots You can find the cube roots and fourth roots, and even higher. The cube root is written as 3 a, the fourth root is written as 4 a. The nth root is therefore written as n a Find the following roots: 1. 3 27 2. 3 125 3. 4 81 4. 4 81