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To solve inequalities, we need to identify whether certain values make the inequality true or false. Inequalities use symbols like >, <, \(eq\), \(\but_not\) represent relationships between expressions, indicating that one expression is either larger, smaller, or not equal to another. For instance, in the linear inequality \(2 x + 3 y > 6\), you are looking to find pairs \( (x, y)\) that make this inequality hold true.
The key steps in solving inequalities generally involve:

• Understanding the inequality and what it represents.
• Substituting the given values (or ordered pairs).
• Performing calculations to verify if the resulting statement is true or false.

Let’s walk through the process with a worked example: The exercise checks if the ordered pairs \( (-3, 3), (5, -1), (0, 2)\) are solutions to the inequality \( 2x + 3y > 6 \). This helps to understand how the values affect the inequality statement. Let's explore more about ordered pairs and their role in inequalities.

Ordered pairs are pairs of numbers, where the order matters. They are usually written in the form \( (x, y)\) where . x represents the value along the x-axis, and y represents the value along the y-axis. In the context of inequalities like \(2 x+3 y>6\).
- Substituting \( (x,y)\) values into the inequality helps us determine if the statement holds true.
In our example, we tested three ordered pairs:

• For \( (-3, 3) \) : Substituting these values into the inequality results in \(2(-3) + 3(3) = 3 \), which is not greater than 6. Therefore, \(-3, 3\) is not a solution.
• For \( (5, -1) \): Substituting these values results in \(2(5) + 3(-1)= 7\), which is greater than 6. Hence, \( (5, -1)\) is a solution.
• Lastly,\((0, 2)\) gives us \( 2(0) + 3(2)= 6 \), which is not greater than 6, thus it is also not a solution.

By understanding and carefully testing ordered pairs in an inequality, you can accurately determine their validity.
Now, let’s learn more about linear inequalities specifically.

Linear inequalities, like linear equations, involve variables raised to the power of one, making a straight-line graph when plotted. However, instead of an equal sign, they use inequality symbols (>,<\(\but_not\), etc.), establishing a range of possible solutions, rather than a single solution.
For example, the linear inequality \( 2 x + 3 y > 6\) includes every point \( (x, y)\) that satisfies this inequality.
To graph such inequalities, follow three general steps:

• Convert the inequality to an equality by replacing the inequality symbol with an equal sign. For \(2 x+3 y>6\), we transform it to \(2 x + 3 y = 6\).This forms a boundary line.
• Plot the boundary line. If the inequality is strict (> or <), use a dashed line. For ≥ or ≤, use a solid line.
• Determine which side of the boundary line represents the solution set. Test a point not on the boundary (usually the origin \( (0,0)\))to see if it fulfills the inequality. If it does, shade the side containing that point; otherwise, shade the opposite side.

Linear inequalities add complexity to problem-solving but understanding the basics makes them manageable. Mastering how to solve and graph them can greatly enhance your algebra skills! Linear inequalities add a fascinating layer of complexity to algebra, making it crucial for developing a robust understanding of mathematical relationships and solution sets.