Problem 65 - Algebra for College Students

Question

Solve each equation and inequality by inspection. Explain how you would solve the inequality $$ |2 x+5|>-3 \text {. } $$

Step-by-Step Solution

Answer

The inequality holds true for all real numbers.

Step 1: Understand the Absolute Value Property

The expression \(|2x + 5|\) represents the absolute value of \(2x + 5\), which means it is the distance of \(2x + 5\) from zero on the number line. Therefore, it is always non-negative. That is, \(|2x + 5| \geq 0\) for all values of \(x\).

Step 2: Analyze the Inequality Comparison

The inequality given is \(|2x + 5| > -3\). We know that any absolute value expression is always \(\geq 0\). Since \(0 > -3\), this inequality holds true for any real number \(x\), because the expression \(|2x + 5|\) will always be greater than -3.

Step 3: Interpret the Solution

Since the inequality \(|2x + 5| > -3\) is always satisfied by any real number due to the property of absolute value expressions being non-negative, we conclude that all real numbers are solutions.

Key Concepts

Inequality Properties

Inequality properties help us understand how to compare different values. They are essential when analyzing mathematical expressions and solving inequalities. A key property to remember is the transitive property, which states: if \(a > b\) and \(b > c\), then \(a > c\). This works similarly for other comparison symbols, such as "less than."
When dealing with absolute values in inequalities, the absolute value is always non-negative, meaning \(|x| \geq 0\) for any real number \(x\). This influences how we approach inequality problems involving absolute values. Since \(|2x + 5| > -3\), we recognize that any absolute value is at least \(0\), which is inherently greater than \(-3\). Thus, any value of \(x\) satisfies this condition.
Understanding these properties allows us to confirm that inequalities remain true across a defined set of values, simplifying the problem-solving process.

Real Numbers

Real numbers are a set of numbers that include both rational and irrational numbers. They can be depicted on a number line and include whole numbers, fractions, and decimal numbers. Real numbers are crucial in inequality solutions as they determine the domain of potential solutions.
The inequality \(|2x + 5| > -3\) invites analysis from the perspective of real numbers. Since the absolute value of any expression is non-negative, every real number is greater than \(-3\). This means all values along the real number line are solutions to the inequality.
It is essential to understand that real numbers encompass a broad range of numeric expressions used in everyday computations, ensuring solutions consider all possible values unless domain restrictions are specified.

Solving Inequalities

Solving inequalities involves finding a range or set of values that satisfy the inequality condition. Unlike equations, inequalities may have multiple or sometimes infinitely many solutions.
For absolute value inequalities like \(|2x + 5| > -3\), the solution step involves recognizing the properties of absolute values: they never take negative values. We compare the absolute value to a number, realizing that, since \(0\) is greater than \(-3\), the inequality holds universally across all \(x\) in the real numbers.
To solve other inequalities, consider manipulating both sides of the inequality to isolate the variable. These manipulations are similar to those used in solving equations, but it's crucial to remember when multiplying or dividing by a negative number, the inequality symbol flips. Keep these techniques in mind when tackling various types of inequalities.