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When dealing with algebraic inequalities, we are often required to translate a verbal statement into a mathematical inequality. This process involves identifying key components and understanding the relationship between these components.
For instance, in the given statement, 'The sum of five times x and twice y is at least 54', we break it down as follows:

• 'Five times x' translates to the expression \( 5x \)
• 'Twice y' translates to the expression \( 2y \)
• 'At least' denotes that the sum must be greater than or equal to 54, which is the inequality symbol \( \geq \)

Combining these pieces, we formulate the inequality as:
\( 5x + 2y \geq 54 \).
This inequality represents all the possible values of x and y that make the statement true. Here, you can plug different values for x and y to see if they satisfy this inequality.

Algebraic expressions are combinations of variables, numbers, and arithmetic operations. In our example, we have two expressions: 'five times x' and 'twice y', represented algebraically as \( 5x \) and \( 2y \), respectively.
An algebraic expression doesn't have an equality or inequality sign; it is simply a part of an equation or inequality. We use these expressions to build equations or inequalities based on the relationships described in a problem statement.
It's important to be comfortable with translating words into algebraic expressions. Here are some common translations:

• 'Sum of' means addition (e.g., sum of a and b is \( a + b \))
• 'Product of' means multiplication (e.g., product of a and b is \( ab \))
• 'Difference of' means subtraction (e.g., difference of a and b is \( a - b \))
• 'Quotient of' means division (e.g., quotient of a and b is \( \frac{a}{b} \))

By practicing these translations, you will become better at forming the correct algebraic expressions for any given situation.

Solving inequalities involves finding the values of the variables that make the inequality true. Let's revisit our example: \( 5x + 2y \geq 54 \).
One way to approach this is to treat it like an equation and try different values for x and y to check if the inequality holds. You can also isolate one variable to solve for the other. Here’s how:

• Subtract one of the terms involving a variable from both sides: \( 2y \geq 54 - 5x \)
• Divide both sides by the coefficient of the remaining variable (if needed) to solve for one variable in terms of the other: \( y \geq \frac{54 - 5x}{2} \)

Once solved, you can plot these solutions on a graph to better understand the range of possible values. Inequalities can have many solutions, represented as a region on the graph rather than just a single point.
Alongside solving inequalities, it's important to remember the properties of inequality:

• When you multiply or divide both sides of an inequality by a negative number, the inequality sign flips (e.g., if \( -a > -b \), then \( a < b \))
• You can add or subtract the same value from both sides without changing the inequality