Problem 4 - Calculus Early Transcendentals: Pearson New International Edition

Question

Express the solution set of the given inequality in interval notation and sketch its graph. $$ 3 x-5<4 x-6 $$

Step-by-Step Solution

Answer

The solution in interval notation is \((1, \infty)\), and the graph is a number line with a hollow circle at 1, shading to the right.

Step 1: Simplify the Inequality

Start by simplifying the given inequality: \[ 3x - 5 < 4x - 6 \] Subtract \(3x\) from both sides to isolate the variable \(x\): \[ -5 < x - 6 \]

Step 2: Solve for x

Next, add 6 to both sides to solve for \(x\): \[ -5 + 6 < x \] \[ 1 < x \] or equivalently, \(x > 1\).

Step 3: Express in Interval Notation

Now, express the solution \(x > 1\) in interval notation. The values of \(x\) begin just after 1 and extend to infinity. Thus, the solution set is: \[ (1, \infty) \]

Step 4: Sketch the Graph

To sketch the graph of the inequality solution, draw a number line. Place a hollow circle on 1 to indicate \(x\) is greater than (but not equal to) 1, and shade the region to the right of 1 to represent all numbers greater than 1.

Key Concepts

Interval Notation

Interval notation is a streamlined way to describe sets of numbers, particularly those defined by inequalities. It offers a concise format to express continuous ranges of numbers. Instead of saying "all numbers greater than 1" in words, interval notation uses a specific shorthand:

• An open parentheses "(" or ")" indicate that a number is not included in the set. For example, \[(1, \infty)\] means the set includes numbers greater than 1 but does not include 1 itself.
• A square bracket "[" or "]" would indicate inclusion of the endpoint. So \([1, 2]\) includes both 1 and 2.
• The symbol \(\infty\) is used to describe numbers that continue indefinitely in a positive direction and "-\infty" for negative directions.

In our solution, \((1, \infty)\) indicates all numbers greater than 1, not including 1, extending infinitely positive. This notation makes it much quicker and clearer to write and understand mathematical solutions.

Solving Inequalities

Solving inequalities is similar to solving equations but with a key distinction: the inequality sign (\(>\), \(<\), \(\geq\), \(\leq\)) indicates a range of possible solutions, not just one fixed value. These types of problems often appear in algebra and serve to develop critical reasoning about numerical relationships.

The process generally involves:

• Combining like terms on each side of the inequality.
• Using basic operations: subtraction or addition, multiplication or division, to isolate the variable on one side.
• Carefully flipping the inequality sign if multiplying or dividing by a negative number.

After isolating the variable, we interpret the sign to determine the range of solutions. In this exercise, for example, simplifying resulted in \(x > 1\). This tells us that any real number greater than 1 satisfies the inequality.

Number Line Representation

Representing solutions graphically on a number line is a powerful visual tool that aids understanding of the range of solutions for inequalities. Here's how you effectively sketch inequality solutions on a number line:

• Start by drawing a straight horizontal line, which represents all real numbers.
• Mark key points, such as where a boundary exists, with numbers along the line.
• For strict inequalities, like \(x > 1\), use an open circle on the number to indicate it's not included in the solution.
• If the endpoint is included (such as \(x \geq 1\)), use a filled circle.
• Shade or draw an arrow in the direction representing all numbers that satisfy the inequality. For \(x > 1\), this would be to the right of 1.

This method provides a clear visual depiction of where the solution lies along the continuum of numbers, helping to solidify comprehension of the inequality's range.