Problem 11 - Algebra and Trigonometry with Analytic Geometry

Question

\(\left\\{\begin{array}{l}3 x+y<3 \\ 4-y<2 x\end{array}\right.\)

Step-by-Step Solution

Answer

The solution is the area between the lines \(y = -3x + 3\) and \(y = -2x + 4\), where \(y < -3x + 3\) and \(y > -2x + 4\).

Step 1: Solve the first inequality for y

The first inequality given is \(3x + y < 3\). To express \(y\) in terms of \(x\), subtract \(3x\) from both sides:\[y < -3x + 3\]

Step 2: Solve the second inequality for y

The second inequality given is \(4 - y < 2x\). To solve for \(y\), subtract 4 from both sides:\[-y < 2x - 4\]Then multiply each term by -1 to solve for \(y\). Remember to reverse the inequality sign when multiplying by negative numbers:\[y > -2x + 4\]

Step 3: Analyze the solution

The solutions to the inequalities are:1. \(y < -3x + 3\)2. \(y > -2x + 4\)This means that the solution set for the system is the region between the lines represented by these inequalities. Since \(y\) must be less than \(-3x + 3\) and greater than \(-2x + 4\), the shaded region where these two conditions overlap on a graph represents the solution.

Key Concepts

Linear Inequalities

Linear inequalities are similar to linear equations but use inequality symbols instead of an equals sign. These inequalities can contain two variables, usually represented by the letters \(x\) and \(y\). Different inequality symbols \(<\), \(>\), \(\leq\), \(\geq\) indicate different relationships:

• \(<\) means less than
• \(>\) means greater than
• \(\leq\) means less than or equal to
• \(\geq\) means greater than or equal to

By rewriting inequalities, like the ones in the exercise, you express one variable in terms of the other. This makes it easier to analyze them using graphs. Instead of finding exact values for \(x\) and \(y\) like in equations, you find a range of values that satisfy the inequality.

Graphical Solution

Graphing is a powerful tool for solving systems of linear inequalities. When you graph an inequality, you begin by graphing the corresponding equation as if the inequality symbol were an equals sign, creating a line on the graph. Decide whether to shade above or below the line based on the inequality symbol. Use a dashed line for \(<\) or \(>\) and a solid line for \(\leq\) or \(\geq\). For instance, if you have \(y < -3x + 3\), you would sketch the line \(y = -3x + 3\) and shade below it, indicating that all points under this line satisfy the inequality.

Solution Set

The solution set is the collection of all possible points that satisfy a system of inequalities. In our case, the solution to the system \(y < -3x + 3\) and \(y > -2x + 4\) is found at the overlap of the shaded regions from each inequality. To see the solution set on a graph:

• Graph both inequalities on the same coordinate plane.
• Identify the region where the shaded areas intersect.
• This overlap is the solution set, meaning all points within this area solve both inequalities.

In our problem, this is a band-like region between the two lines, where \(y\) is less than \(-3x + 3\) and greater than \(-2x + 4\).

Inequalities in Two Variables

Inequalities in two variables are expressions that define a range of values for two different variables. These inequalities divide the plane into two halves. The solution set is one of those halves, determined by whether it satisfies the inequality or not. For example, using \(3x + y < 3\) and \(4 - y < 2x\), after rearranging, you obtain \(y < -3x + 3\) and \(y > -2x + 4\). Each inequality describes a region of the xy-plane. The two-variable nature allows you to graph the inequalities, helping visualize the area where both conditions are met. In practical applications, such regions can represent feasible solutions in optimization problems or constraints in diverse scenarios.