Problem 31 - Beginning and Intermediate Algebra

Question

Solve each inequality. $$ (5 x-1)^{2} \geq 0 $$

Step-by-Step Solution

Answer

All real numbers satisfy the inequality \( (5x-1)^{2} \geq 0 \).

Step 1: Understand the problem

We need to solve the inequality \( (5x-1)^{2} \geq 0 \). Our goal is to find all values of \ x \ that satisfy this inequality.

Step 2: Consider the nature of squares

Remember that the square of any real number \ a^{2} \ is always greater than or equal to 0. This means \( (5x-1)^{2} \geq 0 \).

Step 3: Analyze the inequality

Since \( (5x-1)^{2} \geq 0 \) holds true for any real number, there are no restrictions on \ x \.

Step 4: State the solution

The inequality \( (5x-1)^{2} \geq 0 \) is satisfied for all real numbers \ x \.

Key Concepts

quadratic inequalities

Quadratic inequalities involve expressions where a quadratic polynomial is set in relation to zero using inequalities like \(\geq\), \(\leq\), \(>\), or \(<\). In this context, we dealt with \( (5x-1)^{2} \geq 0\). To solve quadratic inequalities, follow these steps:

• Rewrite the quadratic inequality in standard form, if it's not already.
• Consider the nature of the polynomial – especially the behavior of its roots and square terms.
• Analyze when the expression can be non-negative or non-positive.

Here, knowing that any square term \( a^{2} \) is always greater than or equal to zero helped us conclude the result quickly. Quadratic terms add unique characteristics to inequalities which, once understood, can make these types of problems easier to solve.

real numbers

The concept of real numbers is crucial when solving inequalities. Real numbers include all the numbers on the number line – positive, negative, and zero. This is important because:

• The behavior of functions or inequalities can differ based on whether we're considering the entire set of real numbers or just a subset.
• When solving \( (5x-1)^{2} \geq 0 \), knowing that squares of all real numbers are non-negative simplifies the problem.

In this problem, once we recognized that \( (5x-1) \) is a linear function being squared, we saw it satisfies the condition for all real numbers. Thus, recognizing that real numbers behave uniquely with square functions can offer powerful insights into solving inequalities efficiently.

mathematical reasoning

Mathematical reasoning involves understanding problems deeply and breaking them down logically. When solving inequalities:

• First, correctly interpret what the inequality is asking.
• Identify properties of the functions or expressions involved (like recognizing that a square term is always non-negative).
• Consider the implications of these properties on the potential solutions and restrictions.

In the problem \((5x-1)^{2} \geq 0\), mathematical reasoning led us to analyze the nature of \( a^{2} \) terms. This step-by-step logical approach ensured clarity. By honing these skills, students can tackle inequalities and other mathematical problems more effectively and confidently.