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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. 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.
She starts her video when the bird is feet horizontally from her, and continues filming until the bird is at least 50 feet past her. The bird is flying at a rate of 30 feet per second. Write and solve an equation to find the times after Erin starts filming that the bird is 50 feet horizontally from her.In an absolute value equation, an unknown variable is the input of an absolute value function.
If the absolute value of an expression is set equal to a positive . Solving Absolute-Value Equations and Inequalities Solve absolute-value equations. Solve absolute- and Revise to find values of x that satisfy each absolute-value inequality.
Graph the solution set on a number line. Then use a Write an absolute-value inequality . You can denote absolute value by a pair of vertical lines bracketing the number in question. When you take the absolute value of a number, the result is always positive, even if the number itself is negative.
The absolute number of a number a is written as $$\left | a \right |$$ And represents the distance between a and 0 on a number line.
An absolute value equation is an equation that contains an absolute value expression. This tutorial shows you how to translate a word problem to an absolute value inequality.
Then see how to solve for the answer, write it in set builder notation, and graph it . Absolute Value in Equations and Inequalities 37 SECTION Absolute Value in Equations and Inequalities • Absolute Value and Distance • Absolute Value in Equations and Inequalities • Absolute Value and Radicals This section discusses solving absolute value equations and inequalities.
We start with a geometric deﬁnition of absolute value.