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Chapter 8: Problem 32

Solve each inequality. $$ (4 x+1)^{2} \geq 0 $$

Short Answer

Expert verified
The inequality is satisfied for all real numbers: \(-\infty < x < \infty \).

Step by step solution

01

Understand the inequality

The given inequality is \( (4x + 1)^2 \geq 0 \). We are asked to find values of \(x\) for which this inequality holds true.
02

Analyze the expression \( (4x + 1)^2 \)

Since \((4x + 1)^2\) is a square of a real number, it is always non-negative. That is, \((4x + 1)^2 \geq 0\) for all real numbers \(x\).
03

State the solution

Since the square of a real number is always greater than or equal to zero, the inequality is satisfied for all real numbers. Thus, the solution is \(-\infty < x < \infty \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Inequalities
Inequalities are mathematical expressions showing that one quantity is larger or smaller than another. They use symbols like \( > \), \( < \), \( \geq \), and \( \leq \). When solving inequalities, our goal is to determine the set of values that make the inequality true. The key thing to remember is that inequalities describe a range of possible values rather than a single solution.
For example, in our exercise with the inequality \( (4x + 1)^2 \geq 0 \), we are not looking for one value but for all values of \( x \) that keep the expression non-negative.
This involves analyzing the expression and understanding when and why it remains true or false.
Exploring Real Numbers
The term 'real numbers' refers to all the numbers found on the number line. This includes:
  • Positive numbers
  • Negative numbers
  • Zero
  • Decimals
  • Fractions

Real numbers do not include imaginary numbers, which are based on the imaginary unit \( i \) (where \( i^2 = -1 \)). When solving the given inequality \( (4x + 1)^2 \geq 0 \), we need to consider all real numbers. In this case, any real number substituted for \( x \) in the expression \( (4x + 1) \) will result in a non-negative value when squared. Hence, the inequality holds true for every real number \( x \).
Analyzing Algebraic Expressions
An algebraic expression is a combination of numbers, variables, and mathematical operations. In our exercise, \( (4x + 1)^2 \) is an algebraic expression where:
  • \( 4x \) represents a term with a variable \( x \)
  • \( 1 \) is a constant added to \( 4x \)
  • The whole expression is squared, denoted by \( ^2 \)

The square of any real number results in a non-negative value. This principle simplifies our task because no matter what value \( x \) takes, \( (4x + 1)^2 \) will always be \( \geq 0 \). Hence, solving the inequality simply confirms that all real numbers satisfy it. Understanding and manipulating algebraic expressions are crucial in finding solutions to such inequalities.

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Most popular questions from this chapter

The following exercises are not grouped by type. Solve each equation. $$8 x^{4}+1=11 x^{2}$$

Solve each inequality, and graph the solution set. $$ (x-1)(x-2)(x-4)<0 $$

Solve each equation. Check the solutions. $$4 t=\sqrt{8 t+3}$$

Solve each inequality, and graph the solution set. $$ \frac{6}{x-1}<1 $$

Find the discriminant for each quadratic equation. Use it to tell whether the equation can be solved using the zero-factor property, or the quadratic formula should be used instead. Then solve each equation. (a) \(3 x^{2}+13 x=-12\) (b) \(2 x^{2}+19=14 x\)
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