Which Inequality Is Represented By The Graph Below

To understand which inequality is represented by a given graph, it's important to first recall what an inequality is and how it's visually displayed on a coordinate plane. An inequality is a mathematical statement that compares two expressions using symbols such as < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). When graphed, inequalities show a region of the plane where the statement is true, rather than just a single line or curve.

The first step in identifying an inequality from its graph is to locate the boundary line. This line is often solid if the inequality includes equality (≤ or ≥), or dashed if it does not ( < or > ). For example, if you see a solid line, it means the points on the line are included in the solution set. A dashed line means those points are excluded.

Next, examine which side of the line is shaded. The shaded region represents all the points that satisfy the inequality. To determine which inequality symbol is correct, pick a test point not on the line—often the origin (0,0) is a convenient choice if it's not on the line itself. Substitute the coordinates of the test point into the equation of the line. If the inequality is true for that point, then the region containing that point is the solution set.

For example, if the boundary line is y = 2x + 1 and the region below the line is shaded, you would test the point (0,0): 0 < 2(0) + 1, which simplifies to 0 < 1. Since this is true, the inequality represented is y < 2x + 1. If the line were solid and the region above it shaded, the inequality would be y ≥ 2x + 1.

Sometimes, the graph may show a system of inequalities, where multiple lines and shaded regions overlap. In such cases, the solution is the intersection of all shaded areas. Each line must be analyzed separately for its boundary type and shading direction.

To summarize the process:

1. Identify the equation of the boundary line.
2. Determine if the line is solid or dashed.
3. Choose a test point and check if it satisfies the inequality.
4. Match the shading and boundary type to the correct inequality symbol.

Understanding these steps will help you confidently determine which inequality is represented by any given graph. This skill is essential in algebra and is frequently tested in mathematics courses and standardized exams.