Identifying Linear Inequality From Graph Representation On Coordinate Plane

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In the realm of mathematics, understanding how to represent linear inequalities graphically is a fundamental skill. This article delves into the process of identifying the linear inequality that corresponds to a given graph on a coordinate plane. We will dissect a specific example where a solid straight line with a positive slope intersects particular points, and a shaded region indicates the solution set. By meticulously examining the line's characteristics and the shaded area, we can accurately determine the inequality it represents. This exploration will enhance your comprehension of linear inequalities and their graphical representation, solidifying your mathematical foundation.

Understanding Linear Inequalities

Linear inequalities play a crucial role in various mathematical and real-world applications. They extend the concept of linear equations by introducing inequality symbols, which define a range of solutions rather than a single point. A linear inequality in two variables, typically x and y, can be expressed in several forms, such as y > mx + b, y < mx + b, y ≥ mx + b, or y ≤ mx + b, where m represents the slope and b the y-intercept of the line. The inequality symbol dictates which region of the coordinate plane is included in the solution set.

The graphical representation of a linear inequality provides a visual depiction of all possible solutions. The boundary line, which is the line corresponding to the equation y = mx + b, divides the coordinate plane into two regions. If the inequality includes > or <, the boundary line is dashed to indicate that points on the line are not part of the solution. Conversely, if the inequality includes ≥ or ≤, the boundary line is solid, signifying that points on the line are included in the solution. The shaded region represents all the points (x, y) that satisfy the inequality. Determining the correct inequality from a graph involves identifying the slope and y-intercept of the boundary line, and then using a test point to ascertain which side of the line should be shaded.

Key Components of a Linear Inequality Graph

  • Boundary Line: The boundary line is the graphical representation of the corresponding linear equation. Its slope and y-intercept are crucial in determining the inequality. A solid line indicates that the points on the line are included in the solution (≤ or ≥), while a dashed line indicates they are not (> or <).
  • Slope: The slope (m) of the line determines its steepness and direction. A positive slope means the line rises from left to right, while a negative slope means it falls. The slope can be calculated using two points (x₁, y₁) and (x₂, y₂) on the line with the formula m = (y₂ - y₁) / (x₂ - x₁).
  • Y-intercept: The y-intercept (b) is the point where the line crosses the y-axis. It is the value of y when x = 0. The y-intercept is a key component in the slope-intercept form of a linear equation, y = mx + b.
  • Shaded Region: The shaded region represents all the points (x, y) that satisfy the inequality. The side of the line that is shaded depends on the inequality symbol and the slope of the line. To determine the correct shaded region, a test point (a point not on the line) can be plugged into the inequality. If the point satisfies the inequality, the region containing the point is shaded; otherwise, the other region is shaded.

Analyzing the Given Scenario

In our specific scenario, we are presented with a coordinate plane featuring a solid straight line. This line has a positive slope, indicating that it ascends from left to right. The line passes through two distinct points: (0, -1) and (3, 0). Furthermore, the region above and to the left of the line is shaded. Our objective is to identify the linear inequality that this graph represents.

Determining the Equation of the Line

To determine the equation of the line, we first need to calculate its slope. Given the points (0, -1) and (3, 0), we can use the slope formula:

m = (y₂ - y₁) / (x₂ - x₁) = (0 - (-1)) / (3 - 0) = 1/3

Thus, the slope of the line is 1/3. Next, we identify the y-intercept. Since the line passes through the point (0, -1), the y-intercept is -1. Now we can express the equation of the line in slope-intercept form (y = mx + b):

y = (1/3)x - 1

This equation represents the boundary line of the inequality. Because the line is solid, the inequality will include either ≤ or ≥. The shaded region will help us determine the correct inequality symbol.

Identifying the Correct Inequality

Since the region above and to the left of the line is shaded, we need to determine whether this region corresponds to y ≥ (1/3)x - 1 or y ≤ (1/3)x - 1. To do this, we can use a test point. A convenient test point is (0, 0) because it is easy to substitute into the inequality. Plugging (0, 0) into y ≥ (1/3)x - 1 gives:

0 ≥ (1/3)(0) - 1 0 ≥ -1

This statement is true, which means the region containing (0, 0) should be shaded. Since the shaded region is above the line, this confirms that the inequality is y ≥ (1/3)x - 1.

If we were to test the inequality y ≤ (1/3)x - 1 with the point (0, 0), we would get:

0 ≤ (1/3)(0) - 1 0 ≤ -1

This statement is false, indicating that the region containing (0, 0) should not be shaded, which contradicts the given graph.

Therefore, the correct linear inequality represented by the graph is y ≥ (1/3)x - 1. This meticulous process of determining the slope, y-intercept, and using a test point ensures we accurately identify the inequality.

Step-by-Step Solution

To summarize the step-by-step solution for identifying the linear inequality represented by the graph, we follow a systematic approach that combines algebraic and graphical analysis. This methodical process ensures accuracy and clarity in determining the correct inequality.

  1. Identify Key Features of the Line:

    • Solid or Dashed Line: Determine whether the line is solid or dashed. A solid line indicates that the inequality includes ≤ or ≥, while a dashed line indicates < or >. In our case, the line is solid, so we know the inequality will be either ≤ or ≥.
    • Points on the Line: Note any points through which the line passes. In this scenario, the line passes through (0, -1) and (3, 0). These points will help us calculate the slope and y-intercept.
    • Shaded Region: Observe the shaded region. The area shaded above and to the left of the line provides crucial information about the inequality.

  2. Calculate the Slope:

    • Use the slope formula, m = (y₂ - y₁) / (x₂ - x₁), with the given points (0, -1) and (3, 0).
    • m = (0 - (-1)) / (3 - 0) = 1/3. The slope of the line is 1/3, which is positive, as stated.

  3. Determine the Y-intercept:

    • The y-intercept is the point where the line crosses the y-axis. From the given point (0, -1), we can see that the y-intercept is -1.

  4. Write the Equation of the Line:

    • Use the slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.
    • Substituting m = 1/3 and b = -1, the equation of the line is y = (1/3)x - 1.

  5. Determine the Inequality Symbol:

    • Since the line is solid, the inequality will be either y ≤ (1/3)x - 1 or y ≥ (1/3)x - 1.
    • To determine which inequality is correct, use a test point from the shaded region. A convenient point is (0, 0).

  6. Test the Point:

    • Test point (0, 0) in y ≥ (1/3)x - 1:
      • 0 ≥ (1/3)(0) - 1
      • 0 ≥ -1. This statement is true, so the shaded region should include (0, 0).
    • Test point (0, 0) in y ≤ (1/3)x - 1:
      • 0 ≤ (1/3)(0) - 1
      • 0 ≤ -1. This statement is false, so the shaded region should not include (0, 0).

  7. Select the Correct Inequality:

    • Since the shaded region is above and to the left of the line, and the test point (0, 0) satisfies y ≥ (1/3)x - 1, this is the correct inequality.

Conclusion

In conclusion, the linear inequality represented by the graph is y ≥ (1/3)x - 1. This determination was achieved through a methodical analysis of the line's characteristics, including its slope, y-intercept, and the shaded region. By calculating the slope using the points (0, -1) and (3, 0), identifying the y-intercept as -1, and using the test point (0, 0), we accurately identified the inequality that corresponds to the given graph. Understanding how to interpret and derive linear inequalities from graphs is a critical skill in mathematics, with applications spanning various fields such as economics, engineering, and computer science. This article has provided a comprehensive guide to this process, equipping you with the tools to confidently tackle similar problems and deepen your mathematical acumen.