Problem 46 - A Graphical Approach to College Algebra

Question

Solve each group of equations and inequalities analytically. (a) \(|\pi x+8|=-4\) (b) \(|\pi x+8|<-4\) (c) \(|\pi x+8|>-4\)

Step-by-Step Solution

Answer

(a) No solutions; (b) No solutions; (c) All real numbers.

Step 1: Analyze absolute value equation (a)

The absolute value expression \(|\pi x + 8| = -4\) is evaluated. Since the absolute value of any real number is always non-negative, meaning larger or equal to zero, it is impossible for it to equal \(-4\), a negative number. Thus, there are no solutions.

Step 2: Evaluate absolute value inequality (b)

The inequality \(|\pi x + 8| < -4\) is considered. Absolute values are never negative, so it's impossible for the expression to be less than \(-4\). This means there are also no solutions to this inequality.

Step 3: Solve absolute value inequality (c)

For the inequality \(|\pi x + 8| > -4\), we know that \(|\pi x + 8|\) is always non-negative, hence it will always be greater than any negative value, including \(-4\). Therefore, this inequality is true for all real numbers \(x\).

Key Concepts

Understanding Inequalities

Inequalities are mathematical expressions that describe the relationships between quantities that are not necessarily equal. They use symbols such as \(<\), \(>\), \(\leq\), and \(\geq\). These inequalities can help us understand how numbers relate to one another in terms of size or order. When dealing with absolute values in inequalities, we must think about both positive and negative scenarios.
For example, if we have \(|x| < a\) where \(a\) is a positive number, the solution involves solving \(-a < x < a\). This tells us that \(x\) can be any number within the range of \(-a\) and \(a\).

• Combining absolute values with inequalities often results in compound inequalities.
• To solve them, carefully analyze the "less than" or "greater than" context, taking into account the nature of absolute values always being non-negative.

Exploring Real Numbers

Real numbers encompass all the numbers that can be found on the number line. This includes rational numbers (such as fractions and integers) and irrational numbers (which cannot be expressed as simple fractions, like \(\pi\) and \(\sqrt{2}\)). Real numbers are fundamental in mathematics because they cover every conceivable point on the continuous number line, from infinitely large numbers to infinitesimally small ones.
When solving absolute value equations and inequalities, real numbers play a crucial role.

• Absolute values of real numbers are always non-negative. This is key when analyzing whether equations or inequalities involving absolute values have solutions.
• They allow for expressing solutions to equations in the broadest context possible.

Concept of Non-Negative Values

Non-negative values refer to numbers that are either zero or any positive number. This concept is important when discussing absolute values because an absolute value is defined as the non-negative value of any real number.
For example, the absolute value of \(-3\) written as \(|-3|\) is \(3\), and the absolute value of \(0\) is \(0\). These concepts explain why absolute values can be used to measure distances and why they are always non-negative.
In the context of absolute value inequalities, recognizing that absolute values cannot be negative guides us in understanding why certain inequalities have no solution.

• No real number \(x\) will ever make \(|x|\) less than a negative number.
• Understanding non-negativity ensures that we correctly interpret and solve absolute value problems.