Mr Hernandez Inequality Challenge Finding Solutions On Han's Graph

In a fascinating blend of algebra and graphical analysis, Mr. Hernandez presents a compelling problem to Han, challenging him to deepen his understanding of inequalities and their graphical representations. The core of the problem revolves around plotting points, understanding linear equations, and determining inequalities that satisfy specific conditions. This exploration will not only reinforce key concepts in mathematics but also highlight the interconnectedness of different mathematical ideas.

Plotting the Point (1,1) on Han's Graph

The starting point of this mathematical journey involves Han's graph of the linear equation y = (1/2)x + 2. To plot the point (1,1) on this graph, we must first understand the fundamental nature of linear equations and how they translate into graphical representations. A linear equation, in its simplest form, represents a straight line on a coordinate plane. The equation y = (1/2)x + 2 is in slope-intercept form, where the coefficient of x, which is 1/2, represents the slope of the line, and the constant term, 2, represents the y-intercept. The slope dictates the steepness and direction of the line, while the y-intercept indicates the point where the line crosses the vertical y-axis.

To determine if the point (1,1) lies on the line, we substitute the x and y coordinates of the point into the equation and check if the equation holds true. Substituting x = 1 and y = 1 into the equation y = (1/2)x + 2 yields 1 = (1/2)(1) + 2, which simplifies to 1 = 2.5. This statement is clearly false, indicating that the point (1,1) does not lie on the line represented by the equation y = (1/2)x + 2. This is a crucial observation because it sets the stage for the next part of the problem, which involves finding an inequality that includes this point as a solution. The fact that (1,1) does not lie on the line means it exists in one of the regions defined by the line on the coordinate plane, either above or below it. This spatial relationship is key to understanding how inequalities work graphically.

The process of plotting points and verifying their positions relative to a line is a foundational skill in algebra and coordinate geometry. It underscores the visual nature of mathematical relationships and provides a concrete way to understand abstract concepts. In this case, the act of plotting (1,1) and realizing it does not lie on the line sets the context for exploring inequalities, which are mathematical statements that define regions rather than specific lines or curves.

The Challenge: Adding an Inequality

Mr. Hernandez's challenge to Han is not merely to graph an inequality but to add one that specifically includes the point (1,1) as a solution. This introduces the concept of inequalities and their graphical representation. An inequality, unlike an equation, does not define a single line but rather a region on the coordinate plane. This region consists of all the points whose coordinates satisfy the inequality. For example, an inequality like y > (1/2)x + 2 represents all the points above the line y = (1/2)x + 2, while y < (1/2)x + 2 represents all the points below the line. The inclusion of the “equal to” component, as in y ≥ (1/2)x + 2 or y ≤ (1/2)x + 2, means that the line itself is also included in the solution set.

To determine which inequality Mr. Hernandez could write, we need to test each option by substituting the coordinates of the point (1,1) into the inequality. This process will reveal whether the point satisfies the inequality, meaning it lies within the region defined by the inequality. If the point's coordinates make the inequality a true statement, then that inequality is a possible solution. This method of testing points is a fundamental technique in solving problems involving inequalities and is crucial for understanding the relationship between algebraic expressions and their graphical representations.

The challenge presented by Mr. Hernandez is an excellent way to reinforce the concept that an inequality defines a range of solutions, not just a single solution. It also highlights the importance of understanding how to interpret and manipulate inequalities algebraically to determine their graphical representations. By asking Han to find an inequality that includes a specific point, Mr. Hernandez is encouraging him to think critically about the relationship between points, lines, and regions on the coordinate plane.

Evaluating the Proposed Inequalities

Now, let's methodically evaluate each of the proposed inequalities to determine which one includes the point (1,1) in its solution set. This involves substituting x = 1 and y = 1 into each inequality and checking if the resulting statement is true.

A. y > 2x + 1

Substituting x = 1 and y = 1 into this inequality gives us 1 > 2(1) + 1, which simplifies to 1 > 3. This statement is false. Therefore, the point (1,1) does not satisfy this inequality, meaning it does not lie in the region defined by y > 2x + 1.

B. y < 2x - 1

Substituting x = 1 and y = 1 into this inequality gives us 1 < 2(1) - 1, which simplifies to 1 < 1. This statement is also false. The point (1,1) does not satisfy this inequality either, and thus, it is not a solution for y < 2x - 1.

C. y ≥ -x + 2

Substituting x = 1 and y = 1 into this inequality yields 1 ≥ -(1) + 2, which simplifies to 1 ≥ 1. This statement is true. The point (1,1) does satisfy this inequality, indicating that it lies within the region defined by y ≥ -x + 2. This inequality includes the line y = -x + 2 as well as the region above it.

The process of substituting and evaluating each inequality demonstrates a key method for solving problems involving inequalities. It emphasizes the importance of algebraic manipulation and the ability to interpret mathematical statements accurately. In this case, only one of the inequalities, y ≥ -x + 2, holds true when the coordinates of the point (1,1) are substituted, making it the correct answer. This methodical approach is crucial for building confidence in solving mathematical problems and for developing a deeper understanding of the concepts involved.

Conclusion: The Correct Inequality

Through a careful process of plotting points, understanding linear equations, and evaluating inequalities, we have arrived at the solution to Mr. Hernandez's challenge. The point (1,1) does not lie on the line y = (1/2)x + 2, but it does satisfy the inequality y ≥ -x + 2. This conclusion highlights the importance of understanding the graphical representation of inequalities and how they define regions on the coordinate plane.

This exercise is not just about finding the right answer; it's about reinforcing fundamental mathematical concepts and developing problem-solving skills. By working through this problem, Han, and indeed anyone tackling similar challenges, gains a deeper appreciation for the interconnectedness of algebra and geometry. The ability to translate between algebraic expressions and their graphical representations is a powerful tool in mathematics, and this problem serves as an excellent example of how to cultivate this skill.

Moreover, this problem exemplifies the value of methodical thinking in mathematics. By systematically evaluating each inequality, we were able to confidently identify the correct solution. This approach, which involves breaking down a problem into smaller, manageable steps, is crucial for success in mathematics and in many other areas of life. The exercise reinforces the idea that mathematical problem-solving is not just about memorizing formulas but about understanding concepts and applying them strategically.

In summary, Mr. Hernandez's challenge is a valuable learning experience that reinforces key concepts in algebra and geometry, promotes critical thinking, and highlights the importance of methodical problem-solving. The inequality y ≥ -x + 2 is the correct answer, but the journey to that answer is equally important, providing insights into the nature of mathematical reasoning and the power of visual representation.