Textbook Question
In Exercises 123–124, list all numbers that must be excluded from the domain of each rational expression.3/(2x^2 + 4x - 9)
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1. Equations & Inequalities
The Square Root Property
2:58 minutesProblem 97
Textbook Question
Use the method described in Exercises 83–86, if applicable, and properties ofabsolute value to solve each equation or inequality. (Hint: Exercises 99 and 100 canbe solved by inspection.) | x2 + 1 | - | 2x | = 0
Verified step by step guidance1
Identify the absolute value expressions in the equation: \(|x^2 + 1|\) and \(|2x|\).
Set up the equation by considering the cases where each absolute value expression is non-negative or negative.
For the first case, assume both expressions are non-negative: \(x^2 + 1 = 2x\).
For the second case, assume \(x^2 + 1\) is non-negative and \(2x\) is negative: \(x^2 + 1 = -2x\).
Solve each resulting equation separately to find the possible values of \(x\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Absolute Value
Absolute value measures the distance of a number from zero on the number line, regardless of direction. For any real number x, the absolute value is denoted as |x|, which equals x if x is non-negative and -x if x is negative. Understanding absolute value is crucial for solving equations and inequalities that involve expressions that can take on both positive and negative values.
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Properties of Absolute Value
The properties of absolute value include the fact that |a| = |b| implies a = b or a = -b. This property allows us to set up two separate equations when dealing with absolute values in equations. Additionally, the property |a| - |b| = 0 indicates that the two absolute values must be equal, which is essential for solving the given equation.
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Solving Equations
Solving equations involves finding the values of variables that make the equation true. In the context of absolute value equations, this often requires isolating the absolute value expression and then applying the properties of absolute value to create separate cases. This method is essential for determining all possible solutions to the equation, especially when the equation involves multiple absolute value terms.
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