Problem 53 - Precalculus Student Solutions Manual 5th

Question

Which one of the following inequalities has solution set \((-\infty, \infty) ?\) A. \((x-3)^{2} \geq 0\) B. \((5 x-6)^{2} \leq 0\) C. \((6 x+4)^{2}>0\) D. \((8 x+7)^{2}<0\)

Step-by-Step Solution

Answer

Inequality A, \((x-3)^{2} \geq 0\), has a solution set of \((-fty, fty)\).

Step 1 - Understand the Problem

We are given four inequalities and need to determine which one has a solution set that covers all real numbers \((-fty, fty)\).

Step 2 - Analyze Inequality A

Consider \((x-3)^{2} \geq 0\). The square of any real number is always non-negative. Therefore, this inequality is true for all \x\ in \((-fty, fty)\).

Step 3 - Analyze Inequality B

Consider \((5x-6)^{2} \leq 0\). The square of any real number is always non-negative and is equal to zero only if \(5x - 6 = 0\). Thus, this inequality has a solution set of \(x = \frac{6}{5}\), not all real numbers.

Step 4 - Analyze Inequality C

Consider \((6x+4)^{2} > 0\). The square of any real number is always non-negative and strictly positive except when the expression inside the square is zero. This means the inequality fails at \(6x + 4 = 0\) or \(x = -\frac{2}{3}\), hence not all real numbers.

Step 5 - Analyze Inequality D

Consider \((8x+7)^{2} < 0\). The square of any real number cannot be negative. Therefore, there are no real numbers \x\ that satisfy this inequality.

Step 6 - Conclusion

From the analysis above, only inequality A, \((x-3)^{2} \geq 0\), has a solution set that includes all real numbers \((-fty, fty)\).

Key Concepts

Solution Set

A solution set consists of all values that satisfy a given inequality or equation. In this exercise, we aim to find which inequality's solution set includes all real numbers \((-∞, ∞)\). Here, the concept of a solution set is crucial to understand because it helps determine which values make the inequality true. When analyzing inequalities, it's essential to consider whether the resulting values are valid under the given mathematical operations and constraints. For instance, non-negative squares or strict inequalities both influence the type of solution set we can expect.

Real Numbers

Real numbers include all the numbers on the number line, from negative infinity to positive infinity, including all decimals and fractions. Examples of real numbers are -1, 0, 0.75, and 3.14.
When the problem statement asks us to find a solution set that covers all real numbers \((-fty, ∞)\), it means looking for an inequality true for every possible value within this range.
Whether dealing with non-negative squares or understanding the limitations of inequalities, visualizing real numbers and their properties is fundamental.

Non-Negative Squares

Each core concept—solution set, real numbers, and non-negative squares—sheds light on understanding and solving inequalities effectively. By grasping these concepts, one can easily determine which inequalities hold for all values across the number line.