91Ó°ÊÓ

In algebra, representing unknown values with variables is a fundamental step. By using a variable, like \( x \), we can easily manipulate and solve expressions. Typically, common letters like \( x \), \( y \), or \( z \) are used.
Assigning a suitable variable helps in converting a word problem into a mathematical form. It also lays the groundwork for solving the problem step-by-step. Consider the problem in the exercise: when 5 times a number is subtracted from 6, the result is at least 26. Here, the 'number' is unknown, so we denote it as \( x \).
This quick assignment allows any student to systematically follow through the proceeding steps, such as forming expressions and solving inequalities. It acts as a placeholder for the unknown value that we aim to resolve.

Solving inequalities follows a similar process as solving equations, but with extra care related to the inequality sign. In the given exercise, once the expression \( 6 - 5x \geq 26 \) is set up, the solving process starts with isolating the variable term. This involves a series of logical steps:

• Subtract 6 from both sides: \(-5x \geq 20\).
• Isolate \( x \) by dividing every term with \(-5\).

Here's where the key difference in solving inequalities comes into play. Whenever you multiply or divide both sides of an inequality by a negative number, the inequality sign reverses. Here, dividing by \(-5\) flips \( \geq \) to \( \leq \).
Always confirm your final answer. In this case, the solution \( x \leq -4 \) suggests that the unknown number is any value less than or equal to \(-4\). This verification ensures the proper understanding of both the given problem and its constraints.

Formulating inequality expressions consists of transcribing a word problem into a mathematical expression. This step requires careful attention to linguistic cues describing mathematical operations. For example, in the provided exercise, we identify the phrase _"5 times a number is subtracted from 6"_ and translate it to the expression \( 6 - 5x \).

• "5 times a number" converts to \( 5x \), denoting multiplication.
• "subtracted from 6" guides the expression to \( 6 - 5x \).
• "at least 26" immediately suggests a \( \geq \) inequality.

Thus, the phrase _"at least"_ suggests that the expression joins with a "greater than or equal to" relationship against 26, forming \( 6 - 5x \geq 26 \).
The entire exercise teaches recognizing these indicators to formulate valid expressions, which is critical in the problem-solving of algebraic inequalities. This skill reduces confusion and enhances the accuracy of setting up initial equations needed for further solving.