Problem 93 - Intermediate Algebra

Question

Solve each equation and inequality. For the inequalities, graph the solution set and write it using interval notation. $$ |0.5 x+1|<-23 $$

Step-by-Step Solution

Answer

No solution; an absolute value can never be less than a negative number.

Step 1: Understanding the Absolute Value Inequality

The inequality given is \(|0.5x + 1| < -23\). Before solving, let's note that an absolute value expression such as \(|A|\) represents a non-negative number (always 0 or positive). Therefore, it can never be less than a negative number.

Step 2: Conclusion from Non-Solvable Condition

Since an absolute value \(|0.5x + 1|\) can't be less than a negative number (\(-23\) in this case), there is no solution for this inequality. There are no real values of \(x\) that can make this inequality true.

Key Concepts

Non-Negative Numbers

Understanding non-negative numbers is key when dealing with absolute value inequalities. The absolute value of a number, expressed using vertical bars like \(|A|\), represents its distance from zero on the number line.
This distance is always zero or positive, never negative. For instance, \( |3| = 3\) and \( |-3| = 3\) because both are three units away from zero.
No matter what number or expression is inside the absolute value, its result won't be negative. That's why absolute values are always non-negative numbers.
As a result, when you encounter an inequality that expects a negative value from an absolute expression, like \(|0.5x + 1| < -23\), it presents an impossible scenario.

Inequality Solution

Solving an absolute value inequality involves understanding what the inequality is asking. In this case, \(|0.5x + 1| < -23 \) is impossible to solve.
Absolute values, as mentioned before, cannot be negative, so they cannot be smaller than \(-23\).
When faced with such an inequality, the conclusion is simple: there are no real values of \(x\) that satisfy the equation.
Since you can't satisfy the inequality, you say there is "no solution" for this problem. This makes it a unique type of problem where the process stops at this logical reasoning.

Graphing Solutions

Generally, graphing solutions of inequalities helps visualize the range of possible solutions on a number line. However, in cases where the inequality has no solution, like \(|0.5x + 1| < -23\), there is nothing to graph.
Normally, you would shade the portion of the number line that includes values of \(x\) making the inequality true. But since no values can satisfy the inequality here, the line remains unmarked.
For most problems, graphing highlights the solution set, but this time it serves to confirm that the set is indeed empty.

Interval Notation

Interval notation is a way of writing the solution set of an inequality. Typically, it shows the range of values that make an inequality true using brackets and parentheses.
For example, \( x > 3 \) would be written as \( (3, \infty) \) in interval notation.
Brackets \( [ ] \) are used to include endpoints, while parentheses \( ( ) \) are used to exclude them.
However, for the inequality \(|0.5x + 1| < -23\), which has no solution, we use the notation \( \emptyset \) or simply say "no solution" as there are no values to include in an interval. This communicates the clear absence of a solution set.