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Symbolic manipulation refers to the process of transforming mathematical expressions into a simpler or more useful form. The goal is to isolate the variable we want to solve for. In the context of solving inequalities like \( \frac{x+5}{-10} > 2x + 3 \), symbolic manipulation helps us move each part of the equation until we can clearly see the solution.

Here are a few common steps involved in symbolic manipulation when handling inequalities:

• **Eliminating Fractions**: You might begin by clearing fractions. In our example, multiplying through by \(-10\) removes the fraction. Remember, whenever you multiply or divide by a negative number, **flip the inequality sign**.
• **Distributing Terms**: If there's a number outside of a parenthesis, distribute it across all terms inside. For instance, \(-20\) was distributed to both \(2x\) and \(3\).
• **Combining Like Terms**: Gather all similar terms on the same side. This means putting all \(x\)s together and all constant numbers together.
• **Isolating the Variable**: Work by isolating the variable term. This can be done by adding, subtracting, multiplying, or dividing both sides by the same value.
• **Solving the Inequality**: Finalize by solving for the variable and ensuring the inequality is balanced.

By following these steps, we effectively uncover the range of values for the variable that satisfies the inequality.

Set-builder notation is a method of describing a set by stating the properties its members must satisfy. It provides a clear and concise way to define and express sets. When using set-builder notation, we specify that something belongs to a set, often related to conditions it must meet.

For example, in the inequality solution from earlier, \( x < -\frac{65}{41} \), you could express this range with set-builder notation as:
\[ \{ x \in \mathbb{R} \ | \ x < -\frac{65}{41} \} \]

Here's how to interpret this notation:

• The bracket \(\{ \dots \}\) encloses the description of the elements in the set.
• \(x \in \mathbb{R}\) indicates that \(x\) is an element of the set of all real numbers.
• The condition that \(x\) must satisfy is \(x < -\frac{65}{41}\), which is placed after the vertical bar \(|\).

Using set-builder notation gives a precise mathematical way to convey which numbers belong to the set derived from the inequality solution.

Interval notation is another powerful way to express the solution to an inequality. This method is especially useful for describing continuous ranges of numbers on the number line.

In interval notation, we use parentheses \(( )\) and brackets \([ ]\) to denote intervals. Parentheses are used to indicate that an endpoint is not included, while brackets mean that an endpoint is included.

For our specific inequality \( x < -\frac{65}{41} \), the range of solutions is expressed in interval notation as:
\[ (-\infty, -\frac{65}{41}) \]

Here's a breakdown of what this reflects:

• \((-\infty\) signifies that the interval extends indefinitely to the left on the number line.
• The comma indicates the range from negative infinity up to \(-\frac{65}{41}\).
• \(-\frac{65}{41})\) indicates that \(-\frac{65}{41}\) is not included in the set of solutions.

Interval notation provides a clear and concise way to display sections of the number line that are solutions for inequalities. It allows a fast overview of where solutions are valid without having to express complex conditions.