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Graphing inequalities is a fundamental skill in understanding how different constraints affect possible solutions on a coordinate plane. When you graph an inequality, you visualize all the

• points, or coordinates
• that satisfy the inequality condition.

To graph an inequality like \(x + y > 4\), you first isolate the equation in standard form. Begin by turning the inequality into an equation: \(x + y = 4\). This is the boundary line.
Determine if it should be dashed or solid. In this case, it will be dashed because the inequality does not include equal (\(>\) and not \(\geq\)). This indicates that points on the line do not satisfy the inequality.
Then, choose a test point not on the line to see which side of the boundary includes the points that satisfy the inequality. Usually, (0,0) is a convenient choice, provided it's not on the boundary line. Substitute this point into \(x + y > 4\). If it makes the inequality true, shade that side of the line. If false, shade the opposite side. This shaded area represents the solution set for the inequality.

The solution set of an inequality refers to all possible

• ordered pairs
• or points

that satisfy the inequality's conditions.
For example, using \(x+y>4\) provides a region on the graph where any coordinate pair (x, y) makes the inequality true. Similarly, for \(y<2x-6\), the solution set comprises all points below the line \(y = 2x - 6\).
When determining the union of solution sets, you're combining the areas represented by two or more inequalities.
The union

• includes any point that satisfies at least one of the inequalities,
• expanding the region of valid solutions

without duplicating effort over overlapping regions. This means that even if a point is valid for only one inequality and not the others, it will still be included. The process can often lead to interesting geometric shapes on the coordinate plane, emphasizing the flexibility of linear conditions.

Linear inequalities involve expressions using linear equations with inequality symbols (such as \(>\), \(<\), \(\geq\), or \(\leq\)). These inequalities define a half-plane in a graphical representation. The boundary of this inequality is a straight line, which comes from turning the inequality symbol into an equal sign.
For example, in the inequality \(y < 2x - 6\), the line \(y = 2x - 6\) is the boundary line.
The inequality symbol indicates where shading occurs on the graph:

• either above
• or below the line,

representing the half-plane that satisfies the inequality.
The inequality tells whether to shade above (for \( > \) or \( \geq \)) or below (for \( < \) or \( \leq \)) the boundary line.
This "shaded region" is pivotal because it visually represents all the possible coordinate pairs meeting the condition. An eye-opening part of linear inequalities is when they're combined in a system, showing broader sets of possible solutions.