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Isolating the variable is a fundamental step in solving inequalities and equations. This means we want to get the variable, often represented as \( x \), all by itself on one side of the inequality. The reason for this is simple: once \( x \) is isolated, you have found the solution for \( x \).
To do this effectively, you perform operations that undo whatever is being done to the variable. In our example inequality, \( 8x > 28 \), \( x \) is being multiplied by 8.

To isolate \( x \), divide each side of the inequality by 8. This operation cancels the 8 next to \( x \), leaving you with \( x \) on one side:

• Left side: \( \frac{8x}{8} = x \)
• Right side: \( \frac{28}{8} \)

Remember, whatever operation you perform on one side of the inequality, you must also do to the other side to maintain equality. This ensures that the inequality remains balanced throughout the isolating process.

Once you've isolated the variable, you often need to simplify the fraction you are left with. Simplifying makes the answer more manageable and easier to understand.
In our example, after isolating \( x \), we got the fraction \( \frac{28}{8} \).
To simplify this fraction, find the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 28 and 8 is 4, meaning 28 and 8 can be equally divided by 4 without leaving a remainder.

Here's how you simplify \( \frac{28}{8} \):

• Divide 28 by 4 to get 7.
• Divide 8 by 4 to get 2.

So, \( \frac{28}{8} = \frac{7}{2} \). This simplified form is \( 3.5 \) when expressed as a decimal. Making fractions easier to read helps you and others quickly understand the size of the solution.

Understanding algebraic expressions is key to solving inequalities. An algebraic expression is a combination of numbers, variables, and operations. It represents a statement or equation, made up of terms.
In the inequality \( 8x > 28 \), each part of the expression serves a specific purpose:

• \( 8x \) is a term where 8 is the coefficient, and \( x \) is the variable.
• \( 28 \) is a constant, a number on its own.

Recognizing these components in an expression is crucial to solving any algebraic inequality effectively.
Operations performed on algebraic expressions include addition, subtraction, multiplication, and division, each presenting a unique way to manipulate and analyze equations. By understanding these elements, you'll be better equipped to isolate variables and simplify expressions, helping you solve inequalities with confidence.