Problem 33 - College Algebra with Modeling and Visualization

Question

Solve the inequality. $$ x(x-4) \geq-4 $$

Step-by-Step Solution

Answer

The inequality holds for all real numbers: \( x \in \mathbb{R} \).

Step 1: Set the inequality to zero

First, bring all terms to one side of the inequality to set it to zero. Start with the given inequality: \[ x(x - 4) \, \geq \, -4 \]Add 4 to both sides to obtain:\[ x(x - 4) + 4 \, \geq \, 0 \]

Step 2: Expand and simplify

Expand the expression on the left-hand side:\[ x^2 - 4x + 4 \, \geq \, 0 \]

Step 3: Identify structure and find roots

Notice that the expression \( x^2 - 4x + 4 \) can be recognized as a perfect square trinomial:\[ (x - 2)^2 \, \geq \, 0 \]The expression is zero when \( x - 2 = 0 \), so:\( x = 2 \) is the only root.

Step 4: Analyze the inequality

Since \( (x - 2)^2 \) is a perfect square, it is always non-negative. Therefore, the inequality \( (x - 2)^2 \geq 0 \) holds true for all real numbers. The expression equals zero when \( x = 2 \), but it is greater than zero for all other \( x \).

Step 5: Conclusion

Combine the analysis to conclude that the solution to the inequality is:\[ x \in \mathbb{R} \]This includes all real numbers as the square of any real number is always non-negative.

Key Concepts

Perfect Square Trinomial

Understanding a perfect square trinomial is crucial when dealing with quadratic expressions or inequalities. A perfect square trinomial results from squaring a binomial expression, which means it takes the form:

• \( (a + b)^2 = a^2 + 2ab + b^2 \)
• \( (a - b)^2 = a^2 - 2ab + b^2 \)

In the inequality \( x^2 - 4x + 4 \), we can see that it fits with \( (x - 2)^2 \) because:- Your first term, \( x^2 \), comes from squaring \( x \).- The middle term, \(-4x \), is double the product of \( x \) and \(-2 \).- The last term, \(+4 \), is the square of \(-2 \).This structure helps simplify solving inequalities, as a perfect square trinomial \( (x - 2)^2 \) suggests the expression is always non-negative. Hence, recognizing this type of expression makes it easier to evaluate and solve quadratic inequalities.

Real Numbers

Real numbers are the set of numbers that include all the rational and irrational numbers. These are numbers that we can typically place on an infinite number line.

• Rational Numbers: These are numbers that can be expressed as a fraction, such as \( \frac{1}{2} \), \( -3 \), or 4.
• Irrational Numbers: These cannot be expressed as simple fractions, such as \( \pi \) or \( \sqrt{2} \).

In the context of inequalities, real numbers are important because they allow us to consider values across the entire number line. In the solution provided earlier, the expression \((x - 2)^2 \geq 0\) holds true for all real numbers because the square of a real number is always non-negative. Thus, the solution \( x \in \mathbb{R} \) indicates that any real number will satisfy the inequality.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that represent a certain quantity. They can vary in complexity from simple to extremely intricate expressions. Let's break down some fundamental aspects:

• Variables: These symbols, typically represented by letters such as \( x \), stand in for unknown values.
• Constants: Numbers that have a fixed value, like 4 in the expression \( x^2 - 4x + 4 \).
• Operators: These include addition, subtraction, multiplication, and division, determining how the components interact.

Understanding algebraic expressions helps simplify and solve equations or inequalities. In the inequality given, we start with \( x(x-4) \geq -4 \) and transform it into a more manageable form by expanding and simplifying the terms into \( x^2 - 4x + 4 \). Recognizing this as a perfect square trinomial makes it easier to analyze and solve the inequality.