91Ó°ÊÓ

Algebraic expressions are like phrases in mathematics. They consist of numbers, variables, and operational symbols like addition and multiplication. In algebra, expressions do not have an equal sign; instead, they represent parts of equations or inequalities.
In the inequality we are solving, you notice phrases such as \(7(2x+3)\) and \(5(3x-4)\). These are algebraic expressions that need to be expanded by distributing the multiplication over addition or subtraction within the parentheses.
When expanding, follow these steps:

• Multiply each term inside the parentheses by the number outside. For instance, \(7(2x+3)\) becomes \(14x + 21\).
• Keep track of positive and negative signs when multiplying and distributing terms.

This step is crucial because it simplifies the inequality, making it possible to combine like terms and isolate variables.

Linear inequalities are similar to linear equations, but instead of an equal sign, they use inequality symbols such as \(\leq\), \(\geq\), \(<\), or \(>\). They express relationships where expressions are not equal but rather greater or lesser in value.
For our example, we have the inequality \(18x + 21 \leq 16x - 13\). The solution process involves similar steps as solving linear equations, but you will be focusing on finding a range of possible values for the variable instead of a single value. To solve:

• Bring all variable terms to one side. In this case, subtract \(16x\) from both sides to keep all variations of \(x\) on the left side, leading to \(2x + 21 \leq -13\).
• Continue with solving as you would solve an equation: isolate \(x\) step by step by subtracting constants and then dividing by coefficients.
• Pay attention: if you ever multiply or divide by a negative number, the inequality symbol must be flipped!

Linear inequalities allow us to understand and describe a range of possible solutions rather than a fixed point.

Interval notation is a convenient way of writing down sets of numbers that an inequality covers. It uses brackets and parentheses to precisely define the start and end of an interval.
Here's how intervals are represented:

• Parentheses \(()\) denote that the endpoint is not included.
• Brackets \([]\) show that the endpoint is included.

In the expression \((-\infty, -17]\), this means all numbers less than or equal to \(-17\). The \(-\infty\) represents an unbounded lower end, signifying that there is no actual endpoint on the negative side.
This format is extremely useful in mathematics, especially when communicating solutions to inequalities because it allows a clean and clear presentation of a range of solutions. Make sure to understand the difference between including and not including endpoints, which is often a source of confusion.