Textbook Question
In all exercises, other than exercises with no solution, use interval notation to express solution sets and graph each solution set on a number line.In Exercises 27–50, solve each linear inequality.1 - (x + 3) ≥ 4 - 2x
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1. Equations & Inequalities
Linear Inequalities
3:08 minutesProblem 42
Textbook Question
Explain why the equation | x | = √x² has infinitely many solutions.
Verified step by step guidance1
Understand the definition of absolute value: The absolute value of a number \( x \), denoted as \( |x| \), is the non-negative value of \( x \) without regard to its sign.
Recognize the property of square roots: The square root of \( x^2 \), denoted as \( \sqrt{x^2} \), is also the non-negative value of \( x \).
Compare the expressions: Both \( |x| \) and \( \sqrt{x^2} \) yield the same non-negative value for any real number \( x \).
Conclude that for any real number \( x \), \( |x| = \sqrt{x^2} \) holds true, meaning the equation is satisfied for all real numbers.
Thus, the equation \( |x| = \sqrt{x^2} \) has infinitely many solutions because it is true for every real number \( x \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Absolute Value
The absolute value of a number, denoted as |x|, represents its distance from zero on the number line, regardless of direction. This means |x| is always non-negative. For example, |3| = 3 and |-3| = 3. Understanding absolute value is crucial for solving equations involving it, as it leads to two possible cases based on the sign of x.
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Square Root and Squaring
The square root of a number, represented as √x², is the value that, when multiplied by itself, gives the original number. Since squaring any real number (positive or negative) results in a non-negative value, √x² equals |x|. This relationship is fundamental in understanding how the equation |x| = √x² holds true for all real numbers.
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Infinitely Many Solutions
An equation has infinitely many solutions when there are countless values that satisfy it. In the case of |x| = √x², since both sides are equal for all real numbers x, every real number is a solution. This concept is essential in algebra, as it indicates that the solution set is not limited to discrete values but encompasses a continuous range.
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