Problem 8 - Calculus Early Transcendentals: Pearson New International Edition

Question

Express the solution set of the given inequality in interval notation and sketch its graph. $$ -3<4 x-9<11 $$

Step-by-Step Solution

Answer

The solution set is \((\frac{3}{2}, 5)\).

Step 1: Understand the Compound Inequality

The given inequality is a compound inequality:\[-3 < 4x - 9 < 11\]This means we want values of \(x\) that make both inequalities true: \(-3 < 4x - 9\) and \(4x - 9 < 11\). We'll solve each inequality separately to find these values.

Step 2: Solve the Left Inequality

Solve the inequality \(-3 < 4x - 9\).Add 9 to both sides:\[-3 + 9 < 4x\]\[6 < 4x\]Now, divide both sides by 4:\[\frac{6}{4} < x\]Simplify to:\[\frac{3}{2} < x\]This tells us that \(x\) must be greater than \(\frac{3}{2}\).

Step 3: Solve the Right Inequality

Solve the inequality \(4x - 9 < 11\).Add 9 to both sides:\[4x < 11 + 9\]\[4x < 20\]Divide both sides by 4:\[x < \frac{20}{4}\]Simplify to:\[x < 5\]This tells us that \(x\) must be less than 5.

Step 4: Combine Both Solutions

From Steps 2 and 3, we know:\[\frac{3}{2} < x < 5\]This is the condition for \(x\) to satisfy both inequalities simultaneously.

Step 5: Express Solution in Interval Notation

The solution to the compound inequality in interval notation is:\((\frac{3}{2}, 5)\)This interval describes all \(x\) values strictly greater than \(\frac{3}{2}\) and strictly less than 5.

Step 6: Sketch the Graph

To sketch the graph of the solution set:- Draw a number line.- Mark \(\frac{3}{2}\) and 5 with open circles (indicating these values are not included).- Shade the region between \(\frac{3}{2}\) and 5 to indicate that all values in this interval are part of the solution set.

Key Concepts

Interval Notation

Interval notation is a convenient way to represent a range of values, such as those found in solutions to compound inequalities. By using interval notation, we easily express which values are included or excluded from our solution.
In the inequality \(-3 < 4x - 9 < 11\), after solving, we found that \(\frac{3}{2} < x < 5\). In interval notation, this is written as \(\left( \frac{3}{2}, 5 \right)\), where the parentheses indicate that the endpoints \(\frac{3}{2}\) and \(5\) are not included in the solution set.
This notation is particularly powerful as it succinctly conveys the range without needing words. Always use parentheses for inequalities like > or <, and brackets for ≥ or ≤.

Inequality Graph

An inequality graph provides a visual representation of the solutions to an inequality. It helps to see exactly which values make the inequality true.
For the compound inequality \(-3 < 4x - 9 < 11\), we solve it to find \(\frac{3}{2} < x < 5\), and sketching a graph shows this range clearly.
When graphing, open or closed circles are used to indicate whether endpoints are included in the solution:

• Open circles denote values not included (for < or >).
• Closed circles denote values included (for ≥ or ≤).

The shaded region between the circles indicates all values that satisfy the inequality. This visual form makes understanding and communicating the solution intuitive.

Solving Inequalities

Solving inequalities involves finding all possible values of a variable that make the inequality true. It’s similar to solving equations but requires careful attention to the direction of the inequality sign.
In the example \(-3 < 4x - 9 < 11\), we treat it as two separate inequalities: \(-3 < 4x - 9\) and \(4x - 9 < 11\). Solving them step by step involves simple mathematical operations:

• Addition/Subtraction: Balance the inequality by adding or subtracting the same value on both sides.
• Multiplication/Division: Multiply or divide by a positive number without changing the inequality sign.
• Special Case: If multiplying or dividing by a negative number, the inequality sign is reversed.

The solutions from these steps are then combined to obtain the complete solution.

Number Line Graphing

Using a number line to graph inequalities is a straightforward way to visualize solutions. It helps students identify and understand the range of valid solutions effortlessly.
To represent \(\frac{3}{2} < x < 5\) on a number line:

• Mark the points \(\frac{3}{2}\) and \(5\) with open circles because these endpoints are not part of the solution.
• Shade the region between them to depict valid values for \(x\).

This method is beneficial as it allows easy verification of the solution. If the variable appears in more complex inequalities, graphing can clarify the solution scope and validate each step. Always label your number line for clarity and ease of understanding.