91Ó°ÊÓ

Algebraic expressions are essential components of mathematics, particularly in solving inequalities. An algebraic expression includes numbers, variables (like \(x\)), and arithmetic operations such as addition or multiplication. These expressions form the foundation for equations and inequalities as they allow us to model complex situations mathematically.
In the inequality \(2x + 5 < 19\), \(2x + 5\) is the algebraic expression. It includes:

• A variable term: \(2x\), which means 2 times some number \(x\).
• A constant term: 5, which is added to \(2x\).

Understanding how to read and manipulate these expressions is crucial for solving inequalities. The goal is often to isolate the variable term to determine the values that satisfy the inequality.

Variable isolation is a key strategy in solving inequalities. It involves restructuring the inequality so that the variable stands alone on one side, making it easier to identify solutions.
When dealing with the inequality \(2x + 5 < 19\), the first step is to isolate the term with \(x\). This is achieved by eliminating any constants on the same side as the variable.
To do this, subtract 5 from both sides:

• Original: \(2x + 5 < 19\)
• Subtracting 5 from both sides: \(2x + 5 - 5 < 19 - 5\)
• Simplified: \(2x < 14\)

The next step is to get \(x\) by itself. Divide both sides by 2 to isolate \(x\):

• Division: \(\frac{2x}{2} < \frac{14}{2}\)
• Resulting in: \(x < 7\)

By performing these operations, we've isolated \(x\) and discovered that it represents any value less than 7.

Operations with inequalities are similar to those with equations, but they require extra care in maintaining the inequality relationship. When solving inequalities, you perform arithmetic operations like addition, subtraction, multiplication, and division to both sides.
In this exercise, we performed two main operations:

• Subtraction: This operation was used to eliminate the constant term (5) from the left side of the inequality.
• Division: We divided by 2 to solve for the variable \(x\).

One crucial rule in inequality operations is the behavior of the inequality sign when multiplying or dividing by a negative number. This exercise did not require such action, but it's important to remember:

• If you multiply or divide both sides of an inequality by a negative number, the inequality sign flips direction (\(<\) becomes \(>\) and vice-versa).

These properties ensure that the inequality remains valid as you solve, ultimately leading to the solution \(x < 7\). Understanding these operations helps accurately navigate through more complex inequalities in the future.