Problem 15 - Elementary and Intermediate Algebra

Question

Determine the solution set of each absolute value equation or inequality by inspection. (No work is necessary.) Your answer should be either all real numbers or no solution. \(\begin{array}{lll}{\text { a. }|7 x+6|=-8} & {\text { b. }|7 x+6| \leq-8}\end{array}\) c. \(|7 x+6| \geq-8\)

Step-by-Step Solution

Answer

a. No solution; b. No solution; c. All real numbers.

Step 1: Understand the Concept of Absolute Value

Absolute value represents the distance of a number from zero on the number line and is always non-negative. This means that for any expression \(|A|\), \(|A| \geq 0\). Thus, an absolute value can never be negative.

Step 2: Analyze Equation a

The equation for part a is \(|7x+6| = -8\). Since the absolute value of any expression cannot be negative, there is no value for \(x\) that will satisfy this equation. Thus, the solution set is empty.

Step 3: Analyze Inequality b

The inequality for part b is \(|7x+6| \leq -8\). Again, because an absolute value cannot be less than a negative number, there are no possible solutions that make this inequality true. Hence, there is no solution.

Step 4: Analyze Inequality c

The inequality for part c is \(|7x+6| \geq -8\). Since any absolute value is always greater than or equal to zero, \(|7x+6|\) will always satisfy this inequality no matter the value of \(x\). Therefore, the solution includes all real numbers.

Key Concepts

Understanding the Solution Set in Absolute Value Equations and Inequalities

When solving absolute value equations or inequalities, it's important to know what a solution set is. The solution set is the set of all possible values of the variable that make the equation or inequality true. For a given equation or inequality, you find the solution set by determining which values of the variable satisfy the condition. In the exercises above, you are asked to find these values by inspection. This means you're looking for a solution without doing complex calculations. * If an absolute value equation equals a negative number, like in part a, the solution set is empty because such a condition is impossible. * For an absolute value inequality that is stated to be less than a negative number, similar logic applies, and there is no solution. * Finally, when the inequality states the absolute value is greater than or equal to a negative number, any real number will work. Thus, the solution set is all real numbers. Remember, understanding the nature of absolute values helps in quickly identifying the solution set.

How Inequalities Work in Absolute Value Problems

Inequalities describe a range of values rather than specific numbers. When dealing with absolute values, inequalities can either widen or eliminate the potential solution set.Consider the inequality

• If the absolute value is less than a negative number, there are no possible values that satisfy this because absolute values cannot be negative.
• When the inequality requires the absolute value to be greater than or equal to a number, including negative numbers, every value works since the smallest absolute value can be is zero.

Such understanding allows one to quickly determine the solution set. In exercises like c, where \(|7x+6| \geq -8\) is given, any value will make the inequality true because absolute values are at least zero, offering a valid solution set of all real numbers.

The Role of Real Numbers in Absolute Value Solutions

Real numbers form the backbone of the set of values you consider when solving mathematical problems. They include all rational and irrational numbers on the number line.In absolute value equations and inequalities, especially those like \(|7x+6|\), the variable \(x\) can take on any real number unless restricted by the equation or inequality itself. For example:

• In situations where the condition forces an absolute value equation to equal a positive number, the solutions might be finding specific real numbers.
• If the condition allows the absolute value to be greater than or equal to a non-negative figure, the solution often includes all real numbers, as seen in exercise c.

Therefore, understanding that real numbers are central to finding solutions helps tackle absolute value problems effectively, ensuring that you can define the solution set accurately.