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Inequalities are mathematical expressions that suggest one value is either greater than, less than, or not equal to another. For example, inequalities can look like this:

• \( x > y \) - meaning \( x \) is greater than \( y \)
• \( x < y \) - meaning \( x \) is less than \( y \)
• \( x \geq y \) - meaning \( x \) is either greater than or equal to \( y \)
• \( x \leq y \) - meaning \( x \) is either less than or equal to \( y \)

In solving inequalities, we want to figure out if a particular solution, like a set of values for the variables, makes the inequality true. Unlike equations, inequalities show a range of possible solutions, not just a single one. When solving, it is key to apply basic arithmetic operations consistently to both sides. However, always remember: when multiplying or dividing by a negative number, the inequality sign must be flipped to maintain the true relationship. Checking a solution involves substituting the proposed values into the inequality and confirming the resulting statement is correct.

The substitution method is a strategic approach to solving inequalities or equations by replacing variables with given numerical values. This method simplifies the decision of whether a specific point is a solution of an inequality. To apply this method, follow these simple steps:

• Identify the values for the variable(s), for example, from coordinates of a point like \((x, y)\).
• Substitute these values into each variable in the inequality.
• Solve the resulting expression.

In our example, we used the point \((3, 3)\) in the inequality \(4x - 2y > 6\). By substituting \(x = 3\) and \(y = 3\), we tested if \(4(3) - 2(3) > 6\) was true or false. The aim is to make the left side of the inequality a numerical value to easily compare with the right side.

Evaluating an expression means calculating the expression's value based on given values of variables. This is a crucial step in determining the truth of an inequality. When evaluating:

• Perform multiplication and division operations first, as dictated by the order of operations (often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)).
• For our inequality \(4(3) - 2(3) > 6\), start by calculating the multiplication: \(4 \, \times \, 3\) equals \(12\), and \(2 \, \times \, 3\) equals \(6\).
• Then, subtract the two results: \(12 - 6 = 6\).

Finally, compare this result to the inequality's right side. In this case, you check if \(6 > 6\), which is false, indicating the point \((3, 3)\) does not satisfy the inequality. Evaluating accurately is key to correctly solving and understanding inequalities.