Problem 3 - Calculus Early Transcendentals: Pearson New International Edition

Question

Express the solution set of the given inequality in interval notation and sketch its graph. $$ x-7<2 x-5 $$

Step-by-Step Solution

Answer

The solution set is \((-2, \, \infty)\) and the graph is shaded to the right of -2 with an open dot on -2.

Step 1: Move all terms involving x to one side

Subtract \(x\) from both sides of the inequality to move all terms involving \(x\) to one side of the inequality. This will give us \(-7 < x - 5\).

Step 2: Isolate the variable x

Add 5 to both sides of the inequality to isolate \(x\). This results in \(-7 + 5 < x\), which simplifies to \(-2 < x\).

Step 3: Express in interval notation

The inequality \(-2 < x\) means all numbers greater than \(-2\) are included. In interval notation, this is expressed as \((-2, \, \infty)\).

Step 4: Sketch the graph

On a number line, represent the set of all \(x\) values greater than \(-2\). Use an open dot at \(-2\) to indicate that \(x = -2\) is not included, and shade the line to the right towards infinity, indicating all values greater than \(-2\).

Key Concepts

Interval Notation

When dealing with the solutions to inequalities, interval notation provides a concise way to express ranges of values. Interval notation uses parentheses, square brackets, and infinity symbols to represent the set of numbers that satisfy an inequality.

Here are some key points to remember:

• **Parentheses ( )**: Use these to denote that an endpoint is not included in the interval. For example, \((-2, 3)\) includes all numbers between -2 and 3, but does not include -2 or 3.
• **Square Brackets [ ]**: Indicate that the endpoint is included. For instance, \([-2, 3] \) would include both -2 and 3.
• **Infinity (∞)**: Because infinity and negative infinity aren't numbers, we always use parentheses with them. For example, \((-2, \infty)\) means all numbers greater than -2 are included.

By expressing the solution set in interval notation, we avoid the ambiguity of inequalities and present the solution in easy-to-read terms.

Number Line Graph

A number line graph is a visual representation of a set of numbers, and it is particularly useful for understanding the solution to inequalities. It allows us to see at a glance which numbers are included or excluded from a given set.

To graph an inequality like \(-2 < x\), follow these steps:

• Draw a horizontal line and mark numbers at equal distances, creating a numeric reference.
• Identify the key number or boundary from the inequality, such as -2 in our example.
• Use an open dot at -2 to signify that -2 is not part of the solution.
• Shade the line to the right of -2 to show that all numbers greater than -2 are included.

Number line graphs make it easier to visualize solutions and communicate which values are part of a solution set.

Solving Inequalities

Solving inequalities is similar to solving equations, but with an important difference: the direction of the inequality can change. Here's a general approach to solve linear inequalities:

• **Simplify**: Start by simplifying each side of the inequality if possible.
• **Rearrange**: Move all terms involving the variable to one side of the inequality to isolate the variable.
• **Isolate**: Use addition, subtraction, multiplication, or division to further isolate the variable.
• **Direction**: Remember that multiplying or dividing both sides by a negative number reverses the inequality sign.

Once you solve the inequality, express the solution in interval notation and sketch it on a number line if needed. This gives a clearer understanding of which numbers satisfy the inequality.