Solve each quadratic inequality. Give the solution set in interval notation. (x - 4)² ≤ 0

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Recognize that the inequality is \( (x - 4)^2 \leq 0 \). Since a square of any real number is always non-negative, the expression \( (x - 4)^2 \) is always greater than or equal to zero.

To satisfy the inequality \( (x - 4)^2 \leq 0 \), the expression must be exactly zero because it cannot be negative. So, set the expression equal to zero: \( (x - 4)^2 = 0 \).

Solve the equation \( (x - 4)^2 = 0 \) by taking the square root of both sides, which gives \( x - 4 = 0 \).

Solve for \( x \) by adding 4 to both sides: \( x = 4 \).

Since the inequality holds only when \( x = 4 \), the solution set in interval notation is \( [4, 4] \), which can also be written simply as \( \{4\} \).

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주요 개념

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Quadratic Inequalities

A quadratic inequality involves a quadratic expression set less than, greater than, or equal to a value. Solving it means finding all x-values that satisfy the inequality, often by analyzing the sign of the quadratic expression over intervals determined by its roots.

Perfect Square Trinomials

A perfect square trinomial is an expression like (x - a)^2, which expands to x² - 2ax + a². It is always non-negative because squaring any real number yields zero or a positive result, which simplifies solving inequalities involving such expressions.

Interval Notation

Interval notation is a concise way to represent sets of numbers between two endpoints. It uses parentheses for open intervals (excluding endpoints) and brackets for closed intervals (including endpoints), which is essential for expressing solution sets of inequalities clearly.

Solve each quadratic inequality. Give the solution set in interval notation. (x - 4)2 ≤ 0