Problem 16 - A Graphical Approach to College Algebra

Question

Concept Check \(\quad\) The three-part inequality \(a

Step-by-Step Solution

Answer

Option D cannot be satisfied by any real number.

Step 1: Understanding the concept

The inequality \(a < x < b\) means that the values of \(x\) fall between \(a\) and \(b\). Let's analyze each case provided and check if there is any real number \(x\) that satisfies the conditions.

Step 2: Analyze Option A

For option A (-3 < x < 5), \(x\) can be any real number between -3 and 5, which is possible, as examples like \(x = 0\) or \(x = 2\) satisfy this inequality.

Step 3: Analyze Option B

For option B (0 < x < 4), \(x\) can be any real number between 0 and 4, which is possible, as examples like \(x = 1\) or \(x = 2.5\) satisfy this inequality.

Step 4: Analyze Option C

For option C (-3 < x < -2), \(x\) can be any real number between -3 and -2, which is feasible, as examples like \(x = -2.5\) or \(x = -2.1\) satisfy this inequality.

Step 5: Analyze Option D

For option D (-7 < x < -10), the inequality reads as \(-7 < x < -10\), which is impossible because -10 is less than -7 and there is no real number \(x\) that can be greater than -7 yet less than -10 at the same time.

Key Concepts

Three-part inequalities

Three-part inequalities like \( a < x < b \) represent a range of values that the variable \( x \) can take. It essentially means \( x \) is greater than \( a \) and less than \( b \).
These inequalities are useful in defining intervals on the number line.
For example, the inequality \(-3 < x < 5\) means \( x \) can be any value between \(-3\) and \(5\), excluding \(-3\) and \(5\) themselves.

• You might see these inequalities represented with parentheses \((-3, 5)\).
• The parentheses indicate that the endpoints are not included in the set of possible values for \( x \).

This concept is fundamental for understanding more complex mathematical ideas like absolute values and continuous intervals.

Real numbers

Real numbers encompass all the numbers we typically use in everyday life. They include whole numbers, fractions, and irrational numbers.
When we talk about real numbers in inequalities, we're considering values that exist along a continuous number line.

• Whole numbers, such as \( -1, 0, 2\).
• Fractions, like \( \frac{3}{4} \) or \( -\frac{5}{2} \).
• Irrational numbers, such as \( \sqrt{2} \) and \( \pi \).

Real numbers are crucial in inequalities, as they provide all the potential solutions for an inequality like \( a < x < b \). Thus, if no real number can satisfy a three-part inequality, such as in the case of \(-7 < x < -10\), it's not feasible.

Number line analysis

Number line analysis is a visual way to understand inequalities and the position of values.
By placing numbers and inequalities onto a number line, we can visually interpret the possible values of \( x \).

• Draw a horizontal line and mark significant points, like \( a \) and \( b \).
• Shade the region on the line where \( x \) is valid.
• The open nature of the inequality \( a < x < b \) suggests that points \( a \) and \( b \) are not included, so use open circles to show this.

For instance, the inequality \(-3 < x < 5\) would entail shading the line segment between \(-3\) and \(5\) without including the endpoints.
This technique helps in quickly identifying situations like in option D, where no real number satisfies the inequality because the shaded region is impossible to visualize between \(-7\) and \(-10\).
Number line analysis makes understanding the impossibility of certain inequalities straightforward.