Which of the following inequalities is the solution to the inequality \((x / 2)+3<(x / 3)\) +2 ? A. \(x<-2\) B. \(x<-3\) C. \(x<-6\) D. \(x>-12\) E. \(x<0\)

Inequalities are mathematical expressions that show the relationship between values that are not equal. They use symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to).

Inequalities are fundamental in many areas of mathematics and are used to compare numbers, variables, and other expressions. For example, in the inequality \(x < 5\), the variable \x\ is any number less than 5.

When working with inequalities, remember:

• Addition, subtraction, multiplication, and division can be used to manipulate inequalities, but be careful with multiplication and division by negative numbers as they reverse the inequality sign.
• Use the properties of inequalities to isolate the variable on one side of the inequality.
• Graphing inequalities can provide a visual understanding of the solution set. For a simple linear inequality, this might involve shading a portion of the number line.

Understanding these concepts will help you solve and interpret inequalities efficiently.

Mathematical reasoning involves the ability to analyze and logically work through problems to arrive at a solution. It is essential in solving inequalities as it involves understanding the relationships between different mathematical expressions.

Here’s how you can apply mathematical reasoning to inequalities:

• Identify the inequality: Keep an eye on the inequality sign used and understand what it indicates about the relationship between expressions.
• Transform the inequality: Use algebraic manipulation to rewrite the inequality in a simpler form. This might involve distributing, combining like terms, or eliminating fractions.
• Maintain balance: Ensure any operation performed on one side is also performed on the other side to keep the inequality balanced. Remember, multiplying or dividing by a negative number reverses the inequality.
• Verify solutions: Substitute potential solutions back into the original inequality to ensure they are valid.

Developing strong mathematical reasoning skills will not only help with inequalities but also with a broad range of mathematical problems.

Solving inequalities involves a systematic series of steps to isolate the variable and determine its range. These steps can be broken down as follows:

1. Setup the inequality: Begin by clearly writing down the inequality you need to solve. For example, \(\frac{x}{2} + 3 < \frac{x}{3} + 2\).

2. Eliminate fractions: To work with whole numbers, remove fractions by multiplying every term by the least common multiple (LCM) of the denominators. For our example, multiply through by 6:
\[ 6 \left( \frac{x}{2} + 3 \right) < 6 \left( \frac{x}{3} + 2 \right) \] which simplifies to \[3x + 18 < 2x + 12.\]

3. Isolate the variable: Simplify the inequality by moving all terms containing the variable to one side. Subtract \2x\ from both sides to get:
\[x + 18 < 12.\]

4. Solve for the variable: Perform the necessary arithmetic to isolate the variable. Subtract 18 from both sides:
\[x + 18 - 18 < 12 - 18\]
which simplifies to:
\[x < -6.\]

5. Verify your solution: Compare your solution to the given options. In this case, the solution matches option C. \(x < -6\).

Remember, by following these steps, you can systematically solve inequalities with confidence.