Download Section 2.5 Formulas and Additional Applications from Geometry

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.
Examples:
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
Examples:
Examples:
Solve each inequality and graph the solution set
1. − 1 + 8r < 7r + 2
2. 9 x < −18
3. 7 x − 6 + 1 ≥ 5 x − 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:
if a = 6, then a 2 = 6 2 = 6 ⋅ 6 = 36
If we wish to work backwards, then we would have to take what is called the
square root.
If a 2 = 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
and the process of solving x 2 = 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).
Examples:
1. 16
2. − 169
3.
36
25
4.
− 64
Squaring Radical Expressions:
Squaring a square root removes the radical.
Examples:
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