91Ó°ÊÓ

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols to solve equations and inequalities. In this exercise, we're focusing on inequalities. An inequality has many similarities to an equation, but instead of using the equal sign "=", it uses signs like ">" (greater than) or "<" (less than).
When dealing with inequalities, the main goal is often to isolate the variable on one side of the inequality, much like solving a regular equation. The key rule to remember is that if you multiply or divide both sides of an inequality by a negative number, the inequality sign reverses. However, in our problem, we didn't need to switch signs because we were working with positive coefficients.
To find the solution to an inequality like this one, you will:- Apply algebraic operations like addition, subtraction, multiplication, and division.- Use these operations to isolate the variable, which, in our case, is \(x\).- Remember to apply these operations evenly to both sides of the inequality to maintain its integrity.

A number line is a simple yet powerful visual tool that helps us represent numbers in a linear format. When graphing solutions to inequalities, number lines allow us to visualize which set of numbers satisfies the inequality.
In the given inequality \(x>56\), the solution includes all numbers greater than 56. To graph this on a number line:

• Start by drawing a horizontal line representing real numbers.
• Mark the point for 56 on the line but do not fill it in—it remains an open circle because the number itself is not included in the solution (\(x\) is strictly greater than 56).
• Draw an arrow extending to the right, indicating that all numbers greater than 56 are included in the solution set.

This visual representation makes it clear that any number to the right of 56 on the number line will satisfy the inequality.

Finding solutions to inequalities is important as it tells us what values the variable can take. Once you isolate and solve for the variable, it's essential to check your solution to ensure it's correct.
For the inequality \(2x > 112\), dividing both sides by 2 gives us \(x > 56\). To be confident in this solution, you can choose a value greater than 56, substitute it back into the original inequality \(9(x-11) > 13 + 7x\), and verify the truth of the statement.
Selecting \(x = 57\) as a test value:

• Calculate \(9(57-11) = 9 \times 46 = 414\).
• Calculate the right side: \(13 + 7 \times 57 = 13 + 399 = 412\).
• Since \(414 > 412\), our solution \(x > 56\) indeed holds.

Verification ensures that errors are minimized and that you have a correct and reliable solution for the inequality.