Problem 14 - Algebra for College Students

Question

Solve each inequality. Graph the solution set, and write it using interval notation. \(5 x+6<76\)

Step-by-Step Solution

Answer

x < 14 Interval notation: (-∞, 14)

Step 1 - Isolate the variable

Subtract 6 from both sides to isolate the term with the variable on one side of the inequality: By subtracting 6 from both sides, we get: 5x + 6 - 6 < 76 - 6 This simplifies to: 5x < 70

Step 2 - Solve for the variable

Divide both sides of the inequality by 5 to solve for the variable: Divide by 5: \[ \frac{5x}{5} < \frac{70}{5} \] Simplifying, we get: x < 14

Step 3 - Write the solution in interval notation

The inequality x < 14 describes all numbers less than 14. In interval notation, the solution set is written as: (-∞, 14)

Step 4 - Graph the solution set

To graph the solution set on a number line, draw a number line and shade all parts of the line to the left of 14. Use an open circle at 14 to indicate that 14 is not included in the solution set.

Key Concepts

Inequality

An inequality is a mathematical statement that one quantity is less than, greater than, less than or equal to, or greater than or equal to another quantity. Inequalities use symbols such as <, >, ≤, and ≥. In the given example, the inequality is: 5x + 6 < 76. This inequality tells us that for the value of x, when multiplied by 5 and added to 6, the result should be less than 76.

Variable Isolation

Variable isolation refers to the process of manipulating an equation to have the variable on one side of the inequality or equation. This helps us solve the inequality or equation. Let's go through the steps for our exercise:

In the inequality 5x + 6 < 76, our goal is to isolate x. Here's how you do it:

• Step 1: Subtract 6 from both sides to get: 5x < 70.
• Step 2: Divide both sides by 5 to solve for x: x < 14.

By the end of this process, we have isolated the variable and have a clear solution: x < 14.

Graphical Solution

Graphing an inequality on a number line helps you visualize the solution set. Here's how you can graph x < 14:

• Draw a horizontal number line.
• Mark the value 14 on the number line.
• Use an open circle at 14 to indicate that 14 is not included in the solution.
• Shade the number line to the left of 14 to show all numbers less than 14.

By following these steps, you create a clear visual representation of the solution, making it easier to understand which values of x satisfy the inequality.

Interval Notation

Interval notation is a way of writing subsets of the real number line. It helps in describing the solution set of inequalities. Enclosing numbers in parentheses ( ) indicates that the endpoint is not included, while square brackets [ ] indicate that the endpoint is included. For the inequality x < 14:

• The solution set is all numbers less than 14.
• In interval notation, we write this as: (-∞, 14).

The parentheses around -∞ and 14 indicate that neither endpoint is included in the set. This concise notation is very helpful for representing ranges of values.