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When solving inequalities or equations, one of the first steps you might encounter is the need to apply the distributive property. This property helps us simplify expressions by distributing a single term across terms inside parentheses. Here鈥檚 how it works:
Suppose you have the expression: \(5(t-3) + 4t\).
To use the distributive property, multiply 5 by each term inside the parentheses:
\[ 5(t-3) = 5t - 15 \]
We do the same on the other side of the inequality:
\(2(7+2t)\) becomes \[ 2 \times 7 + 2 \times 2t = 14 + 4t \]
This step is crucial because it allows us to combine like terms and isolate the variable effectively. Understanding and correctly applying the distributive property ensures a smoother path to solving the inequality.

Combining like terms is another important step in simplifying mathematical expressions. Like terms are terms that contain the same variable raised to the same power. To simplify:
Look at the left side of our inequality: \(5t - 15 + 4t\).
Here, \(5t\) and \(4t\) are like terms. Adding them gives:
\[ (5t + 4t) - 15 = 9t - 15 \]
Combining like terms reduces the complexity of the expression, making it easier to isolate the variable. This simplification is key to solving the inequality efficiently and correctly.

Set-builder notation is a method for describing a set by stating the properties that its members must satisfy. After solving the inequality, we write the solution in set-builder notation:
We found that \(t < 5.8\).
Thus, in set-builder notation, the solution is written as:
\[ \{ t | t < 5.8 \} \]
This reads as: 鈥淭he set of all \(t\) such that \(t\) is less than 5.8.鈥�
Set-builder notation provides a concise way to represent an infinite set of solutions that satisfy a given condition, which is particularly useful in inequality solutions.

Interval notation is another way to represent sets of numbers, emphasizing the range of values that satisfy an inequality. For the solution \(t < 5.8\), it is written in interval notation as:
\[(-\infty, 5.8)\]
This notation uses parentheses and brackets to denote which endpoints are included in the set:

• Round brackets \((\) indicate that an endpoint is not included in the interval.
• Square brackets \([\) would indicate an endpoint is included, but in this case, with \(t < 5.8\), 5.8 is not included.

Interval notation provides a clear, visual way to represent the span of solutions across the number line, making it an efficient tool for conveying solution sets.