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Inequality manipulation involves performing operations on both sides of an inequality to isolate the variable or simplify the expression. These operations must maintain the inequality's direction unless multiplying or dividing by a negative number.

For the original exercise, the first step involved subtracting 3t from both sides to help isolate the variable 't'. After this subtraction, you end up with a simpler inequality: \[2t - 3 \geq 10\]

Next, we added 3 to both sides to remove the constant term on the left, resulting in \[2t \geq 13\].

Finally, we divided both sides by 2 to solve for 't', giving us the final solution: \[t \geq \frac{13}{2}\].

This step-by-step manipulation ensures the inequality remains valid.

Algebraic expressions are combinations of variables, numbers, and operations. They can represent relationships and quantities in mathematical problems.

In our example, the expression [5t - 3] represents a quantity that must be greater than or equal to another expression [3t + 10]. Breaking down these expressions helps simplify and solve the inequality.

When manipulating these expressions, we use basic arithmetic operations (addition, subtraction, multiplication, and division) to simplify them step by step. Each simplification brings us closer to isolating the variable.

Variables in inequalities represent unknown quantities that we aim to solve for. When solving inequalities, our goal is to isolate the variable on one side of the inequality.

In the exercise provided, the variable 't' represents the unknown quantity. Our objective is to find the range of values that 't' can take while still satisfying the inequality.

We carefully manipulated the inequality to isolate 't' and determine that \[t \geq \frac{13}{2}\]. This means that any value of 't' that is greater than or equal to \[\frac{13}{2}\] will satisfy the original inequality.

Understanding how to work with variables in inequalities allows for solving more complex problems and interpreting mathematical relationships accurately.