Problem 22 - Intermediate Algebra

Question

Solve each inequality. Graph the solution set and write it using interval notation. See Example 2. $$ x-5>2 $$

Step-by-Step Solution

Answer

The solution is \( x > 7 \), shown as \( (7, \infty) \).

Step 1: Isolate the variable

To solve the inequality, begin by isolating the variable \(x\) on one side of the inequality. Given the inequality \(x - 5 > 2\), add 5 to both sides to eliminate the constant on the left side. \[ x - 5 + 5 > 2 + 5 \] Simplify both sides to get \[ x > 7 \]

Step 2: Graph the solution set

To graph \(x > 7\), draw a number line. Locate the point corresponding to 7 on the number line. Since the inequality is strictly greater than 7, draw an open circle at 7 to indicate that 7 is not included in the solution set. Then, draw a line extending to the right to represent all numbers greater than 7.

Step 3: Write the solution in interval notation

The solution in interval notation includes all numbers greater than 7 and excludes 7 itself. Therefore, write the solution in interval notation as \( (7, \, \infty) \). This indicates that the solution set includes all real numbers greater than 7, extending indefinitely.

Key Concepts

Solving Inequalities

Inequalities are like equations, but instead of equality signs, they use signs like \( < \), \( > \), \( \leq \), and \( \geq \). Solving an inequality involves finding all the values of the variable that make the inequality true.
To solve the inequality \( x - 5 > 2 \), you start by isolating the variable \( x \). This involves manipulating the inequality just like you would an equation. The goal is to get \( x \) by itself on one side. Here, you would add 5 to both sides:

• \( x - 5 + 5 > 2 + 5 \)

This simplifies to \( x > 7 \). This means that any number greater than 7 will satisfy the inequality, because when you substitute it back into the original inequality, the statement remains true.
It's important to remember that if you multiply or divide both sides of an inequality by a negative number, the inequality sign flips direction.In this problem, however, we only added a number, so the inequality sign stays the same.

Graphing Solutions

Graphing the solution to an inequality like \( x > 7 \) helps to visualize the set of solutions.
First, draw a number line which is a horizontal line that represents numbers.
Locate the point 7 on this line.

• Because the inequality is strictly \( > 7 \) and does not include 7, you use an open circle at 7. This open circle shows that 7 is not part of the solution.
• Then draw a line or arrow extending to the right from the open circle. This line indicates that all numbers greater than 7 are included in the solution. The arrow suggests that it continues indefinitely.

This graphical representation offers a quick way to see the range of values that satisfy the inequality. It also highlights that the inequality has infinitely many solutions to the right of 7.

Interval Notation

Interval notation is a shorthand way of writing the solution to inequalities. It is especially useful because it is compact and easy to read.
For the inequality \( x > 7 \), we want to show all numbers greater than 7 but not including 7 itself.

• This is written in interval notation as \( (7, \infty) \). The parenthesis "(" at 7 indicates that 7 is not included in the set, which corresponds to the open circle used in graphing.
• The symbol \( \infty \) is used to indicate that the solutions continue infinitely. Infinity is always paired with an open end (parenthesis) because it is not a number that can be reached.

Interval notation provides a clear, concise, and standardized way to express solutions to inequalities, making communication of mathematical ideas more efficient.