Decoding Linear Inequality Graphs A Step-by-Step Solution

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Understanding linear inequalities and their graphical representation is a fundamental concept in mathematics. This article will guide you through the process of identifying a linear inequality from its graph on a coordinate plane. We will dissect a specific example, where a dashed straight line with a positive slope passes through the points (-1, -1) and (0, 1), with the region to the left of the line shaded. Our goal is to determine the linear inequality that corresponds to this graphical representation. Before diving into the specifics of the problem, let's establish a firm understanding of the core concepts involved. First and foremost, it's vital to grasp what a linear inequality actually is. In essence, it's a mathematical statement that compares two expressions using inequality symbols, such as < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). These inequalities, when graphed on a coordinate plane, represent a region rather than a single line, differentiating them from linear equations, which depict straight lines.

In this particular scenario, we are presented with a dashed line. This is a crucial piece of information. A dashed line indicates that the points on the line itself are not included in the solution set of the inequality. This is because the inequality will involve either a strict "less than" (<) or a strict "greater than" (>) symbol. Conversely, a solid line would signify that the points on the line are part of the solution, implying the use of a "less than or equal to" (≤) or "greater than or equal to" (≥) symbol. The fact that the line is dashed immediately narrows down our options, eliminating any inequalities with ≤ or ≥. Another critical element to analyze is the shaded region. The shaded area represents all the points (x, y) that satisfy the inequality. In our case, the region to the left of the dashed line is shaded. This provides valuable insight into the type of inequality we are dealing with. If the region above the line were shaded, it would typically suggest a "greater than" inequality, while shading below the line usually implies a "less than" inequality. However, since we are dealing with the region to the left of the line, and the line has a positive slope, this suggests a "greater than" inequality. This is because, for a positive slope, the y-values increase as we move to the right. Therefore, points to the left will have smaller y-values relative to the line.

To determine the linear inequality, we first need to find the equation of the line. The equation of a line can be expressed in slope-intercept form as y = mx + b, where 'm' represents the slope and 'b' represents the y-intercept. We are given two points that lie on the line: (-1, -1) and (0, 1). We can use these points to calculate the slope 'm' using the formula: m = (y2 - y1) / (x2 - x1). Plugging in the coordinates, we get: m = (1 - (-1)) / (0 - (-1)) = (1 + 1) / (0 + 1) = 2 / 1 = 2. Therefore, the slope of the line is 2. Next, we need to find the y-intercept 'b'. The y-intercept is the point where the line crosses the y-axis, which occurs when x = 0. We are conveniently given the point (0, 1), which directly tells us that the y-intercept is 1. Thus, b = 1. Now that we have both the slope (m = 2) and the y-intercept (b = 1), we can write the equation of the line in slope-intercept form: y = 2x + 1. However, remember that we are looking for a linear inequality, not an equation. We need to incorporate the information about the dashed line and the shaded region to determine the correct inequality symbol.

As discussed earlier, the dashed line indicates that the points on the line are not included in the solution. This means we will use either > or <. The shaded region to the left of the line, combined with the positive slope, suggests that we should use the > symbol. This is because, for a given x-value, the y-values in the shaded region are greater than the corresponding y-value on the line. To confirm this, let's consider a point in the shaded region, say (-2, 0). If we substitute these coordinates into the inequality y > 2x + 1, we get: 0 > 2(-2) + 1 which simplifies to 0 > -4 + 1 or 0 > -3. This statement is true, confirming that the point (-2, 0) satisfies the inequality. Now, let's test a point on the other side of the line, say (1, 3). Substituting these coordinates into y > 2x + 1, we get: 3 > 2(1) + 1 which simplifies to 3 > 2 + 1 or 3 > 3. This statement is false, as 3 is not greater than 3. This further confirms that the correct inequality should only include the shaded region to the left of the line. Therefore, the linear inequality represented by the graph is y > 2x + 1. This means that any point (x, y) that satisfies this inequality will lie in the shaded region, while points on the line and to the right of the line will not satisfy the inequality.

Now that we have determined the linear inequality represented by the graph, we can compare it to the given answer choices. The answer choices provided are:

A) y > 2x + 2 B) y ≥ 1/2x + 1 C) y > 2x + 1 D) y ≥

By comparing our derived inequality, y > 2x + 1, to the answer choices, we can clearly see that option C) y > 2x + 1 matches our result. Let's briefly analyze why the other options are incorrect.

Option A) y > 2x + 2 has the correct inequality symbol but a different y-intercept. The y-intercept in this inequality is 2, while the y-intercept of the line in the graph is 1. Therefore, this option is incorrect.

Option B) y ≥ 1/2x + 1 has an incorrect inequality symbol (≥ instead of >) and an incorrect slope (1/2 instead of 2). The solid line indicated by ≥ also contradicts the dashed line in the graph. Therefore, this option is incorrect.

Option D) y ≥ is incomplete and doesn't provide a valid linear inequality. Therefore, this option is incorrect.

In conclusion, the linear inequality represented by the graph is y > 2x + 1. This exercise highlights the crucial steps involved in analyzing linear inequality graphs:

  1. Identifying the type of line (dashed or solid) to determine the inequality symbol (> or < vs. ≥ or ≤).
  2. Analyzing the shaded region to understand which side of the line satisfies the inequality.
  3. Calculating the slope and y-intercept to find the equation of the line.
  4. Combining these elements to construct the linear inequality.

By mastering these steps, you can confidently interpret and solve problems involving linear inequalities and their graphical representations. Understanding these concepts is not only essential for academic success in mathematics but also for various real-world applications where linear relationships and constraints need to be analyzed.

  • Linear equation
  • Slope-intercept form
  • Inequality symbols
  • Graphing linear inequalities
  • Solution set
  • Coordinate plane

By exploring these keywords further, you can deepen your understanding of linear inequalities and related mathematical concepts, paving the way for more advanced topics in algebra and calculus.