Problem 10 - Calculus Early Transcendentals: Pearson New International Edition

Question

Express the solution set of the given inequality in interval notation and sketch its graph. $$ 4<5-3 x<7 $$

Step-by-Step Solution

Answer

The solution set is \((-\frac{2}{3}, \frac{1}{3})\).

Step 1: Break Down the Compound Inequality

The given compound inequality is \(4 < 5 - 3x < 7\). We'll treat this as two separate inequalities: \(4 < 5 - 3x\) and \(5 - 3x < 7\). We'll solve each inequality separately before combining the solution.

Step 2: Solve the First Inequality

Start with the inequality \(4 < 5 - 3x\). Subtract 5 from both sides to isolate \(-3x\):\[ 4 - 5 < -3x \]. This simplifies to \(-1 < -3x\). Divide both sides by -3, remembering to reverse the inequality sign: \[ \frac{-1}{-3} > x \], which simplifies to \( \frac{1}{3} > x \) or \( x < \frac{1}{3} \).

Step 3: Solve the Second Inequality

Now solve the second inequality \(5 - 3x < 7\). Subtract 5 from both sides to get \(-3x < 2\). Then, divide both sides by -3, reversing the inequality sign: \[ x > \frac{2}{-3} \], which simplifies to \( x > -\frac{2}{3} \).

Step 4: Combine the Solutions

The solutions for the inequalities are \( x < \frac{1}{3} \) and \( x > -\frac{2}{3} \). These combine to form the compound inequality \(-\frac{2}{3} < x < \frac{1}{3} \).

Step 5: Express in Interval Notation

The solution set in interval notation can be written as \( \left(-\frac{2}{3}, \frac{1}{3}\right) \). This represents all values of \(x\) between \(-\frac{2}{3}\) and \(\frac{1}{3}\), not including the endpoints.

Step 6: Sketch the Graph

To graph this interval, draw a number line and mark the points \(-\frac{2}{3}\) and \(\frac{1}{3}\). Use open circles at these points since they are not included in the solution. Draw a line between these two points to indicate all values \(x\) such that \(-\frac{2}{3} < x < \frac{1}{3}\).

Key Concepts

Compound Inequalities

Compound inequalities are like two separate inequalities joined together by the word 'and' or 'or'. These are solved by finding the values of the variable that satisfy both inequalities at once. For example, the compound inequality \(4 < 5 - 3x < 7\) can be split into two parts: \(4 < 5 - 3x\) and \(5 - 3x < 7\). Each part defines a range of values for \(x\), and together they create an overlapping solution set. This overlap is the solution to the compound inequality. By working through the inequality step-by-step, you can find the range of values that satisfies both qualities, which is crucial in giving a complete solution.

Interval Notation

Interval notation is a way of writing a set of numbers, often used to describe the solution set of inequalities. It uses brackets and parentheses to indicate whether endpoints are included or excluded. For example, in the solution \((-\frac{2}{3}, \frac{1}{3})\), the parentheses signify that the endpoints \(-\frac{2}{3}\) and \(\frac{1}{3}\) are not included in the solution.

• Round brackets \(( )\) mean the endpoint is not included.
• Square brackets \([ ]\) mean the endpoint is included.

In our inequality, since the values do not include the endpoints, parentheses are used. This method provides a clean and quick way to express solutions to inequalities, useful for both writing and understanding them easily.

Number Line Graph

Graphing inequalities on a number line helps visualize solution sets. To create a number line graph for the compound inequality \(-\frac{2}{3} < x < \frac{1}{3}\), you begin by drawing a straight line and marking the critical values

• \(-\frac{2}{3}\) and \(\frac{1}{3}\) are marked as open circles because these points are not part of the solution.

Between these points, you draw a line to show all numbers in this range are part of the solution. This visual representation makes the concept of inequalities more tangible, highlighting the continuous nature of solutions between these points.

Inequality Solution Methods

There are several methods to solve inequalities, which often differ slightly from solving equations. A key point when solving inequalities is that multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign. For example, in solving \(-3x < 2\), dividing both sides by \(-3\) requires flipping the less than sign to greater than sign \(x > -\frac{2}{3}\).

• Always isolate the variable on one side.
• Use inverse operations like addition, subtraction, multiplication, or division.
• Reverse the inequality sign when multiplying or dividing by a negative number.

Using these methods correctly ensures accurate solutions, essential in solving compound inequalities, as seen in the original exercise. Understanding and applying these basic principles allows you to efficiently find valid solutions to inequalities.