Textbook Question
Solve each quadratic inequality. Give the solution set in interval notation. (x - 4)2 ≤ 0
Ch. 3 - Polynomial and Rational Functions
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
Chapter 4, Problem 16
Solve each quadratic inequality. Give the solution set in interval notation. -(x + 1)2 ≥ 0
Verified step by step guidance1
Rewrite the inequality clearly: \(-(x + 1)^2 \geq 0\).
Recognize that \((x + 1)^2\) is a perfect square and is always greater than or equal to zero for all real \(x\).
Multiply by the negative sign outside the square, which makes \(-(x + 1)^2\) less than or equal to zero, since the square is nonnegative and the negative sign flips the inequality direction.
Set the expression equal to zero to find critical points: \(-(x + 1)^2 = 0\) which simplifies to \((x + 1)^2 = 0\).
Solve \((x + 1)^2 = 0\) to find \(x = -1\). Use this to determine where the inequality holds and express the solution set in interval notation.
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1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Quadratic Inequalities
A quadratic inequality involves a quadratic expression set greater than or less than a value, such as ≥ 0. Solving it requires finding the values of the variable that make the inequality true, often by analyzing the sign of the quadratic expression.
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Nonlinear Inequalities
Properties of Squares and Non-Positivity
Since squares of real numbers are always non-negative, expressions like -(x + 1)^2 are always less than or equal to zero. Understanding this helps determine when the inequality holds, especially recognizing when the expression equals zero or is negative.
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Interval Notation
Interval notation is a concise way to express solution sets of inequalities using intervals and endpoints. It uses parentheses for open intervals and brackets for closed intervals, indicating whether endpoints are included or excluded.
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