Problem 76 - College Algebra and Calculus: An Applied Approach

Question

\(y\) is at least six units from \(0 .$$x\) is less than eight units from \(0 .\)

Step-by-Step Solution

Answer

The solution set to these inequalities does not exist as there is no common value that satisfies both conditions simultaneously.

Step 1: Translate the Expressions

The expression 'is at least six units from 0' corresponds to the inequality \(|y| \geq 6\) and 'is less than eight units from 0' corresponds to the inequality \(|x| < 8\) in terms of absolute values.

Step 2: Translate to Standard Inequalities

Translate the absolute value inequalities to standard inequalities. For \(y\), it gives \(y > 6\) or \(y < -6\) and for \(x\), \(x < 8\) and \(x > -8\).

Step 3: Finding the solutions

Examine the inequalities to find any common solutions. Here \(y\)'s solution can be any real number greater than 6 or less than -6. Similarly, \(x\)'s solution can be all real numbers between -8 and 8. Thus, the common solution is not possible as there is no intersection. Therefore, no solution exists.

Key Concepts

Inequality Translation

In mathematics, translating verbal expressions into inequalities is the first step to solving problems involving constraints. When you read a statement like "at least six units from 0," it indicates a distance from 0 in a number line context. The concept of absolute value comes into play here, where distance can only be non-negative.

The phrase "is at least six units from 0" translates into the inequality \(|y| \geq 6\). This expression implies that the value of \(y\) can be further than 6 in either the positive or negative direction from 0. Similarly, "less than eight units from 0" is represented by the inequality \(|x| < 8\). This tells us that the possible values for \(x\) must lie within a certain range, specifically between -8 and 8.

Translating these expressions correctly is crucial because it lays the foundation for finding the number solutions and any intersections.

Real Number Solutions

Once the inequality translation is clear, we solve for the possible values of variables within the constraints. Looking at the absolute value inequality \(|y| \geq 6\), we're essentially looking for all real numbers further away from 0 than 6.

This breaks down into two separate inequalities to find the real number solutions:

• \(y > 6\) which captures all numbers greater than 6.
• \(y < -6\) which captures all numbers less than -6.

Similarly, for \(|x| < 8\), we need to consider numbers within a specific zone around zero:

• \(x < 8\) and also,
• \(x > -8\)

Together, this tells us that the solution set for \(x\) includes all real numbers between -8 and 8, providing a bounded segment of the number line. Whereas the solution set for \(y\) extends indefinitely away from two points—on both ends of the number line.

No Common Intersection

Once real number solutions are identified, checking for common solutions becomes the next step. Intersections occur where the solutions to two inequalities overlap, meaning both can be true at the same time.

For the problem at hand, however, we examine two separate solutions without overlap for \(y\) and \(x\):

• The \(y\) solution, including all numbers greater than 6 or less than -6, essentially creates two separate, endless intervals.
• The \(x\) solution, constrained between -8 and 8, exists entirely within a portion of the number line.

Since the allowed values for \(x\) do not reach or overlap the conditions required for \(y\), there's simply no intersection. These solution sets are said to have 'no common intersection', making it impossible for both inequalities to be satisfied simultaneously by any real number. This leads us to the conclusion that no real number can satisfy both inequalities at the same time, leaving us with no solution for the system of inequalities.