Problem 45 - A Graphical Approach to College Algebra

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: Identify Solution for Absolute Equation

For the equation \(|5 - 7x| = 0\), we need to solve for \(x\) when the absolute value expression equals zero. This occurs when the expression inside the absolute value, \(5 - 7x\), equals zero.

Step 2: Solve for x

Set the expression inside the absolute value equal to zero: \(5 - 7x = 0\). To solve for \(x\), add \(7x\) to both sides: \(5 = 7x\). Then, divide both sides by 7: \(x = \frac{5}{7}\).

Step 3: Interpret Result for Inequalities

For \(|5 - 7x| \geq 0\), note that an absolute value is always non-negative, so this inequality is true for all real numbers \(x\). For \(|5 - 7x| \leq 0\), the only time the absolute value is zero is when \(x = \frac{5}{7}\), as found earlier.

Key Concepts

Solving Inequalities

In mathematics, solving inequalities involves finding the set of all possible values that satisfy a given inequality. Unlike equations, inequalities deal with ranges rather than specific solutions. When solving an inequality, it's crucial to understand the direction of the inequality sign which can be one of the following:

• Greater than (">")
• Less than ("<")
• Greater than or equal to ("≥")
• Less than or equal to ("≤")

Inequalities can sometimes involve absolute values, which add another layer of consideration due to the non-negative nature of absolute values.
For example, consider \(|5 - 7x| \geq 0\). Since the absolute value is always non-negative by definition, this inequality is automatically satisfied for all real numbers. Conversely, \(|5 - 7x| \leq 0\) describes a condition where the absolute value can only be zero. Thus, this inequality is true only when the expression inside the absolute value itself is zero, making the solution specific and not a range.

Algebraic Expressions

Algebraic expressions form the foundation of all algebra. They are combinations of numbers, variables, and arithmetic operations. An algebraic expression might look like \(5 - 7x\), where "5" is a constant, "x" is a variable, and "-7" is the coefficient of "x".

• **Constants** are fixed values.
• **Variables** represent unknown values.
• **Coefficients** are numbers multiplying the variables.

When dealing with algebraic expressions in equations or inequalities, it's important to manipulate them to isolate the variable. This allows you to solve for the variable's value. For instance, if you have \(5 - 7x = 0\), the goal is to get "x" on one side. By rearranging, we set \(5 = 7x\) and subsequently divide by 7, giving \(x = \frac{5}{7}\).
Understanding how to rearrange and simplify these expressions is key in tackling more complex problems.

Real Numbers

Real numbers encompass all the numbers we encounter in everyday life, both rational and irrational. They are an essential part of mathematics because they form the continuous number line we use to measure and represent quantities.

• **Rational Numbers** include integers, fractions, and finite decimals.
• **Irrational Numbers** include non-repeating, non-terminating decimals such as \(\pi\) and \(\sqrt{2}\).

When dealing with the inequalities \(|5 - 7x| \geq 0\), it becomes clear that the solutions span all real numbers. This is because an expression involving an absolute value greater than or equal to zero has no restrictions except in cases where the inequality would result in an impossible condition (in this case, impossible conditions do not exist since absolute values are inherently non-negative).
Recognizing that solutions to inequalities can be any real number or a specific subset is critical in algebra. It also highlights the versatility and breadth of real numbers as a mathematical set.