Question

With two minutes left in a basketball game, your team is 12 points behind. What are two different numbers of 2 -point and 3-point shots your team could score to earn at least 12 points? Sketch the graph of the inequality. Then name two ways your team could score at least 12 points.

Short answer

Two combinations to score at least 12 points are: 1) Making six 2-point shots (total 12 points) or 2) Making four 3-point shots (total 12 points).

Step-by-step solution

Identify variables

Let's denote the 2-point shots as \(x\) and 3-point shots as \(y\). We need to find the values of \(x\) and \(y\) such that \(2x + 3y \geq 12\). This inequality represents the fact that the team needs to make a combination of 2-point and 3-point shots that total a minimum of 12 points.

Graph the inequality

To graph the inequality \(2x + 3y \geq 12\), firstly, we need to consider the equation of a line, \(2x + 3y = 12\), which divides the plane into two parts. By choosing several points on the line (e.g., \((0,4)\) and \((6,0)\)), we can draw the line on the Cartesian plane. This line is the boundary of the two areas. To decide which side of the line corresponds to the solution of the inequality, we can use test points. Since \(0 \leq x, y\), we will test the origin \((0,0)\). As it doesn't satisfy the inequality, the other side of the line is the solution area. Therefore, all the points in that area and the points on the line are the solution of the inequality.

Identify solutions

Now, we can find at least two solutions which satisfy the inequality \(2x + 3y \geq 12\). Note that \(x\) and \(y\) must be non-negative integers as they represent the number of shots. We could select \((x,y)\) as \((6,0)\), i.e., six 2-point shots or we could select \((x,y)\) as \((0,4)\), i.e., four 3-point shots.

Key concepts

Systems of Inequalities

Understanding systems of inequalities is a foundational skill in algebra that helps address questions about multiple conditions that need to be satisfied simultaneously. An inequality, similar to an equation, is a statement regarding the relative size or order of two expressions. However, an inequality doesn't indicate exactness but rather a range of possible solutions. In the context of our basketball game problem, where the team needs at least 12 points to tie or win, we are working with a single inequality, but this concept is often expanded into systems.

A system of inequalities would consist of multiple inequalities that represent different conditions. The graphical solution for such systems involves finding the area of intersection where all the inequalities overlap. This area represents all the possible solutions that satisfy the entire system. In real-world problems, this could represent constraints like budgets, resource limitations, or even scoring goals in sports, as in the textbook example. The process involves plotting each inequality and then identifying the overlapping region, which introduces interdisciplinary skills that combine both algebraic problem-solving and understanding the Cartesian coordinate system.

Algebraic Problem-Solving

The heart of algebraic problem-solving lies in developing a methodical approach to tackling mathematical challenges. The process involves identifying variables and writing equations or inequalities that represent the real-world situation. In our exercise, we've defined the number of 2-point shots as variable \(x\) and the number of 3-point shots as variable \(y\). After forming an inequality \(2x + 3y \geq 12\), we can explore the various combinations of \(x\) and \(y\) that satisfy the condition.

Key steps in algebraic problem-solving typically include defining variables, forming equations, solving for variables, and interpreting results. It's crucial to ensure the solutions make sense in the context of the problem. In the given scenario, for instance, we must remember that we can't have fractional shots; hence, we seek only whole number solutions. Problem-solving in algebra often entails trial and error with test points, analyzing graphs, and strategic thinking about the most effective methods to reach a solution.

Cartesian Coordinate System

The Cartesian coordinate system is a vital mathematical construct for graphing equations and inequalities. It consists of two perpendicular number lines, the x-axis (horizontal) and the y-axis (vertical), that intersect at the origin (0,0). The system allows us to visually represent relationships between variables as points on a plane. Graphing inequalities involves plotting a line that represents the boundary of the region where the inequality is true.

In our basketball example, the line \(2x + 3y = 12\) was drawn, with points such as \((0,4)\) and \((6,0)\) lying on it. The entire half-plane that includes points where \(2x + 3y\) is greater than or equal to 12 represents the solution to the inequality. By using this graphical method, students can quickly see all the possible combinations of 2-point and 3-point shots that meet the game's criteria. It provides an intuitive and visual way to understand and solve inequalities, which is an invaluable tool in learning and applying algebraic concepts.