Problem 41 - Algebra 2

Question

Which One Doesn't Belong? Given the following system of inequalities, which ordered pair does not belong? Explain your reasoning. $$y \leq \frac{1}{2} x+5 \quad y<-3 x+7 \quad y \geq-\frac{1}{3} x-2$$

Step-by-Step Solution

Answer

The ordered pair that does not belong is the one that doesn't satisfy all three inequalities.

Step 1: Interpret the System of Inequalities

The system of inequalities consists of three linear inequalities that define regions in a coordinate plane. The inequalities are: 1) \( y \leq \frac{1}{2}x + 5 \) 2) \( y < -3x + 7 \) 3) \( y \geq -\frac{1}{3}x - 2 \). We are asked to determine which point, if any, doesn't belong to the region defined by this system.

Step 2: Identify the Intersection Region

To find the region described by the system, we solve for the areas where all inequalities overlap in the coordinate plane. 1) \( y \leq \frac{1}{2}x + 5 \) represents the area below the line \( y = \frac{1}{2}x + 5 \).2) \( y < -3x + 7 \) represents the area below the line \( y = -3x + 7 \). 3) \( y \geq -\frac{1}{3}x - 2 \) represents the area above the line \( y = -\frac{1}{3}x - 2 \). The solution region is where all these conditions meet on the graph.

Step 3: Check Each Given Ordered Pair

Assume we are given the ordered pairs to be checked. Let's check which ones satisfy all three inequalities:1. For each order pair \((x, y)\), substitute the \(x\) and \(y\) values into each inequality.2. Verify if all inequalities hold true for each pair.3. Identify any pair that does not satisfy all three inequalities.

Key Concepts

linear inequalities

Linear inequalities involve expressions with a variable raised to the power of one. They can take forms such as \( y \leq \frac{1}{2}x + 5 \) or \( y > -3x + 7 \). These inequalities describe relationships where one side is not precisely equal but either less than, less than or equal to, greater than, or greater than or equal to the other side.
Understanding linear inequalities is essential as they define sets of possible solutions. Unlike equations that have fixed answers, inequalities allow for a range of solutions. They graph as half-planes on a coordinate plane, providing a visual representation of all possible solutions.
When graphing linear inequalities, it's important to consider the following:

• Inequality signs determine whether to use a solid line (for \( \leq \) or \( \geq \)) or a dashed line (for \( < \) or \( > \)).
• The inequality indicates which side of the line should be shaded to represent the solution area.
• Test points can help confirm which region satisfies the inequality.

coordinate plane

The coordinate plane is a two-dimensional space typically defined by an x-axis (horizontal) and a y-axis (vertical). It is used to graph equations and inequalities which helps visualize the solutions at a glance.
In a coordinate system, every point is represented by an ordered pair \((x, y)\), where \(x\) is the point's horizontal position, and \(y\) is its vertical position. This grid-like system allows us to locate points, plot lines, and identify regions easily.
When solving a system of inequalities, the coordinate plane helps in finding the intersection region by illustrating where the solutions to each inequality meet. Graphing the equations on the same plane, we look for overlapping shaded regions which represent all points satisfying the entire set of inequalities. This graphical approach allows complex systems to be interpreted more straightforwardly.

intersection region

The intersection region on a coordinate plane is where the solutions of multiple inequalities overlap. When dealing with systems of inequalities, this region represents the set of points that satisfy all inequalities simultaneously.
To determine the intersection region:

• Start by graphing each inequality on the same coordinate plane.
• Observe the common shaded areas across all inequalities; this is the intersection region.
• Verify the boundaries defined by each inequality to ensure accuracy of the solution region.

Finding the intersection region is crucial in answering questions like, "Which point doesn't belong?" It involves checking whether each point lies within the intersection of all solution regions. Any point outside of this region will not satisfy the system as a whole.