Problem 57 - A Graphical Approach to Precalculus with Limits

Question

Solve each group of equations and inequalities analytically. (a) \(|5-7 x|=0\) (b) \(|5-7 x| \geq 0\) (c) \(|5-7 x| \leq 0\)

Step-by-Step Solution

Answer

(a) \(x = \frac{5}{7}\); (b) all real numbers; (c) \(x = \frac{5}{7}\).

Step 1: Solve Absolute Value Equal to Zero

For the equation \(|5 - 7x| = 0\), the absolute value is zero if the expression inside is zero itself. So, we set up the equation:\[5 - 7x = 0\]Solve for \(x\) by adding 7x to both sides:\[5 = 7x\]Then divide both sides by 7:\[x = \frac{5}{7}\]Thus, the solution to the equation \(|5 - 7x| = 0\) is \(x = \frac{5}{7}\).

Step 2: Solve Absolute Value Greater Than or Equal to Zero

For the inequality \(|5 - 7x| \geq 0\), recognize that the absolute value of any real number is always non-negative. Therefore, the inequality holds true for all real values of \(x\).So, the solution is all real numbers.

Step 3: Solve Absolute Value Less Than or Equal to Zero

For the inequality \(|5 - 7x| \leq 0\), the only time the absolute value is zero is when the expression inside is zero. From Step 1, we know that this only happens when \(x = \frac{5}{7}\). So, the solution is \(x = \frac{5}{7}\).

Key Concepts

Inequalities

Inequalities are a fundamental concept in mathematics that compare two expressions relative to each other. Unlike equations, which state that two expressions are equal, inequalities show a range of possible solutions. There are several types of inequalities, such as:

• Greater than (\(>\)
• Less than (\(<\)
• Greater than or equal to (\(\geq\)
• Less than or equal to (\(\leq\)

The problem in the example involves absolute value inequalities. An absolute value represents the distance of a number from zero on a number line, which is always non-negative. In this context, an inequality like \(|5-7x| \geq 0\) is true for all real numbers because distance cannot be negative. By understanding the nature of absolute values, the range of solutions for inequalities becomes easier to identify. Moreover, recognizing such characteristics aids in solving a variety of mathematical problems.

Analytical Solutions

Analytical solutions involve resolving mathematical problems using algebraic manipulations and logical reasoning to find exact solutions. This approach is fundamental for solving equations and inequalities accurately. Let’s break it down further:

• Identify the nature of the problem: Recognize whether you are dealing with an equation or an inequality.
• Use algebraic techniques: Apply operations like addition, subtraction, multiplication, division, and factoring to isolate variables.
• Logical reasoning: Analyze the conditions under which the solution holds, especially in cases involving absolute values.

For the equation \(|5-7x| = 0\), solving analytically meant setting the expression inside the absolute value equal to zero, as its absolute value is zero only when the expression itself is zero. This logical reasoning leads to an exact solution: \(x = \frac{5}{7}\). Analyzing problems this way allows one to derive solutions based on the definitions and properties of numbers within mathematical structures.

Real Numbers

In mathematics, real numbers encompass all numbers that can be represented on the number line, including rational and irrational numbers. They form the foundation of many mathematical concepts and problems. Understanding real numbers involves recognizing that:

• Rational numbers: Numbers that can be expressed as the quotient of two integers, such as \(\frac{5}{7}\).
• Irrational numbers: Numbers that cannot be written as a simple fraction, like \(\pi\) and \(\sqrt{2}\).
• They include integers, decimals, and fractions.

In the context of this exercise, the inequality \(|5-7x| \geq 0\) was solved by acknowledging that absolute values are always non-negative, meaning any real number satisfies this condition. Therefore, the solution, "all real numbers," applies because any value of \(x\) will make the inequality true. Understanding real numbers enhances our ability to comprehend a wide range of mathematical problems and their solutions.