Solve each quadratic inequality. Give the solution set in interval notation. See Example 1. (x - 4)^2 ≤ 0

Verified step by step guidance

1. 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.
2. Understand that the only way for \((x - 4)^2\) to be less than or equal to zero is if \((x - 4)^2 = 0\) exactly, because squares cannot be negative.
3. Set the expression inside the square equal to zero: \(x - 4 = 0\).
4. Solve for \(x\) by adding 4 to both sides: \(x = 4\).
5. Conclude that the solution set is the single value \(x = 4\), which in interval notation is written as \([4, 4]\) or simply \(\{4\}\).

Key Concepts

Quadratic Inequalities

A quadratic inequality involves a quadratic expression set to be greater than, less than, or equal to a value. Solving it requires finding the values of the variable that satisfy the inequality, often by analyzing the sign of the quadratic expression.

Perfect Square Trinomials

A perfect square trinomial is an expression like (x - a)^2, which expands to x^2 - 2ax + a^2. Recognizing this form helps simplify solving inequalities, especially when the expression is squared and set less than or equal to zero.

Interval Notation

Interval notation is a way to express the set of solutions to inequalities using intervals. It uses parentheses for open intervals and brackets for closed intervals, clearly indicating which values are included or excluded in the solution set.