High School Statutory Authority: Algebra I, Adopted One Credit.
The absolute value is always positive, so you can think of it as the distance from 0. I like to then make the expression on the right hand side without the variables both positive and negative and split the equation that way. In this case our answer is all real numbers, since an absolute value is always positive.
Here are more problems: Try the answers in the original equation to make sure they work! Note that we still have to simplify first to separate the absolute value from the rest of the numbers. Check the answers; the work!
Note that we get some complex roots since we had to take the square root of a negative number. We need to treat the absolute value like a variable, and get it out from the denominator by cross multiplying.
Then we can continue to solve, and divide up the equations to get the two answers.
Section Linear Systems with Two Variables. A linear system of two equations with two variables is any system that can be written in the form. Note: Although systems of linear equations can have 3 or more equations,we are going to refer to the most common case--a stem with exactly 2 lines. Case I: 1 Solution This is the most common situation and it involves lines that intersect exactly 1 time. Preface. Some time ago I searched for a textbook for a sophomore course in differential equations that would combine analytical (algebraic) methods of solution with graphical and numerical methods in .
Check the answers — they work! In this case, we have to separate in four cases, just to be sure we cover all the possibilities.
Solving Absolute Value Inequalities When dealing with absolute values and inequalities just like with absolute value equationswe have to separate the equation into two different ones, if there are any variables inside the absolute value bars.
We first have to get the absolute value all by itself on the left. Now we have to separate the equations. We get the first equation by just taking away the absolute value sign away on the left.
The easiest way to get the second equation is to take the absolute value sign away on the left, and do two things on the right: Here are some examples: Even with the absolute value, we can set each factor to 0, so we get —4 and 1 as critical values.
Then we check each interval with random points to see if the factored form of the quadratic is positive or negative, making sure we include the absolute value.
Then we need to get everything to the left side to have 0 on the right first. Simplify with a common denominator. We see the solution is: Graphs of Absolute Value Functions Note that you can put absolute values in your Graphing Calculator and even graph them!
Absolute Value functions typically look like a V upside down if the absolute value is negativewhere the point at the V is called the vertex. Applications of Absolute Value Functions Absolute Value Functions are in many applications, especially in those involving V-shaped paths and margin of errors, or tolerances.
Problem Solution Two students are bouncing-passing a ball between them. Create an absolute value equation to represent the situation. How high the did the ball bounce for the second student to catch it? Suppose that a coordinate grid is placed over a putt-putt golf course, where Amy is playing golf.
Write an equation for the path of the ball.
Here are examples that are absolute value inequality applications: This makes sense since anything outside of these values would be more than.
And we also know the difference of the temperature and 72 has to be in this range. Therefore, we can write this temperature range as an absolute value and solve: A bird is approaching Erin, a photographer, and she films it.In mathematics, a recurrence relation is an equation that recursively defines a sequence or multidimensional array of values, once one or more initial terms are given: each further term of the sequence or array is defined as a function of the preceding terms..
The term difference equation sometimes (and for the purposes of this article) refers to a specific type of recurrence relation. Preface. Some time ago I searched for a textbook for a sophomore course in differential equations that would combine analytical (algebraic) methods of solution with graphical and numerical methods in .
(Note that with non-linear equations, there will most likely be more than one intersection; an example of how to get more than one solution via the Graphing Calculator can be found in the Exponents and Radicals in Algebra section.) Solving Systems with Substitution.
A system of nonlinear equations is two or more equations, at least one of which is not a linear equation, that are being solved simultaneously. A system of linear equations that has at least one solution is called.., whereas a system of linear equations that has no solution is called..
consistent, inconsistent. The process used to write a system of linear equation in row-echelon form is . A System of Linear Equations is when we have two or more linear equations working together.