Problem 65 - College Algebra with Modeling and Visualization

Question

Solve the compound linear inequality graphically. Write the solution set in set-builder or interval notation, and approximate endpoints to the nearest tenth whenever appropriate. $$ x-4<2 x-5<6 $$

Step-by-Step Solution

Answer

The solution set in interval notation is \((1, 5.5)\).

Step 1: Break Down the Compound Inequality

The given compound inequality is \( x - 4 < 2x - 5 < 6 \). Break it down into two separate inequalities. The first is \( x - 4 < 2x - 5 \), and the second is \( 2x - 5 < 6 \).

Step 2: Solve the First Inequality

Solve \( x - 4 < 2x - 5 \). Subtract \( x \) from both sides to get \( -4 < x - 5 \). Next, add 5 to both sides, resulting in \( 1 < x \).

Step 3: Solve the Second Inequality

Solve \( 2x - 5 < 6 \). Add 5 to both sides to get \( 2x < 11 \). Divide both sides by 2 to solve for \( x \), resulting in \( x < 5.5 \).

Step 4: Combine the Solutions

Combine the results from the two inequalities. The solution is where both conditions are true simultaneously: \( 1 < x < 5.5 \).

Step 5: Write the Solution in Interval Notation

The combined solution can be written in interval notation as \( (1, 5.5) \). This represents all \( x \) values between 1 and 5.5, exclusive.

Key Concepts

Interval Notation

Interval notation is a concise way to represent a set of numbers that lie within a certain range on a number line. It uses parentheses and brackets to indicate whether endpoints are included or excluded.
In our solution, we arrived at the inequality \( 1 < x < 5.5 \). To express this in interval notation, we'd write \( (1, 5.5) \).
This notation means:

• The round brackets \( (\) and \()) \) tell us that 1 and 5.5 are not part of the solution set.
• If the inequality were \(1 \leq x < 5.5\), we'd change the \(1\) to a square bracket \([\), showing that 1 is included.
• Interval notation offers a clear, shorthand way to describe which numbers lie between (but not including) those specific points.

Interval notation is very useful in mathematics as it gives a compact form of writing and visualizing solutions.

Set-Builder Notation

Set-builder notation is another technique for describing a set, particularly useful when the set is defined by a property that its members must satisfy.
When using set-builder notation, you'd describe the solution set of our inequality \( 1 < x < 5.5 \) as:
\[ \{ x \mid 1 < x < 5.5 \} \]
Here's what this means:

• The curly braces \( \{ \} \) signify a set of numbers.
• The vertical bar \( \mid \) is read as "such that" and is used to separate the elements of the set from the condition they satisfy.
• In this case, "\( x \mid 1 < x < 5.5 \)" translates to "the set of all \( x \) such that \( x \) is greater than 1 and less than 5.5."

Set-builder notation is especially helpful when you need to involve more complex conditions or when working within a particular context — offering flexibility to include expressions and additional constraints.

Graphical Solutions

Graphical solutions involve plotting the solution of an inequality on a number line or coordinate plane, offering a visual interpretation of the solution set.
For the combined inequality \( 1 < x < 5.5 \), graphing on a number line can be done as follows:

• Draw a horizontal line which will act as your number line.
• Mark the points '1' and '5.5' on this line.
• Use open circles (or parentheses, depending on your tool) at these points, indicating that these endpoints are not included in the solution set.
• Shade the region between \( x = 1 \) and \( x = 5.5 \), as this is the range where \( x \) is valid.

This method helps visually demonstrate the impact of each part of the inequality and makes it easier to communicate complex ideas. It can also be especially beneficial for students who are visual learners, helping them grasp the range and limitations of \( x \) directly from the graph.