Problem 26 - Thinking Mathematically

Question

Graph the solution set of each system of inequalities. \(\left\\{\begin{array}{c}x+y<4 \\ 4 x-2 y<6\end{array}\right.\)

Step-by-Step Solution

Answer

The solution of the system of inequalities is the region where the shaded areas of the inequalities \( y < -x + 4 \) and \( y > 2x - 3 \) overlap.

Step 1 - Graph First Inequality

Begin by graphing the first inequality \( x + y < 4 \). This can be rewritten as \( y < -x + 4 \). The graph of this will be a line with a slope of -1 and y-intercept of 4. Because it is less than \( < \), the area below the line is shaded to represent possible solutions.

Step 2 - Graph Second Inequality

Graph the second inequality \( 4x - 2y < 6 \). This can be rewritten as \( y > 2x - 3 \). The graph of this will be a line with a slope of 2 and y-intercept of -3. Since it is greater than \( > \), the area above the line is shaded to represent possible solutions.

Step 3 - Identify Solution Area

Finally, identify the region where the shaded areas from both inequalities overlap. This region is the solution of the system of inequalities as it satisfies both conditions. Label this region clearly.

Key Concepts

System of Inequalities

A system of inequalities consists of multiple inequalities that are considered simultaneously. In this exercise, we are working with two inequalities:

• \( x + y < 4 \)
• \( 4x - 2y < 6 \)

These inequalities share the same variables, \( x \) and \( y \). Therefore, the solution must satisfy **both** inequalities at the same time. Solving such a system involves finding all the ordered pairs \( (x, y) \) that make both inequalities true. Once we find this set of points, we identify it graphically as a common shaded region, representing the solution set.

Solution Set

The solution set of a system of inequalities is the region on a graph where all the conditions given by the inequalities are satisfied. To find the solution set, we first plot each inequality on the coordinate plane:

• For \( x + y < 4 \), all points below the line \( y = -x + 4 \) are valid.
• For \( 4x - 2y < 6 \), the points above the line \( y = 2x - 3 \) are valid.

The solution set is where the shaded areas of these two inequalities overlap. This intersection represents all \( (x, y) \) values that simultaneously fulfill the system of inequalities. It helps us to visualize and comprehend the range of possible solutions on a graph.

Linear Inequalities

Linear inequalities are similar to linear equations but use inequality signs \( <, >, \leq, \geq \) instead of an equal sign. In graphical terms, a **linear inequality** divides the coordinate plane into two parts:

• One part where the inequality holds true.
• The other part where it does not.

Let's examine our specific inequalities:

• \( x + y < 4 \) implies that \( y \) should be less than \( -x + 4 \).
• \( 4x - 2y < 6 \) suggests that \( y \) should be more than \( 2x - 3 \).

Each inequality automatically directs us to shade either above or below the line to represent the region of valid solutions. Linear inequalities provide a way to visualize constraints and determine viable solutions through graphing.

Graphical Representation

Graphical representation is crucial in solving a system of inequalities as it provides a visual means to determine the solution set. For our exercise:

• We draw the line \( y = -x + 4 \) for the inequality \( x + y < 4 \), shading the area below for the solution.
• We then graph \( y = 2x - 3 \) for \( 4x - 2y < 6 \) and shade above the line.

The intersection of these shaded areas is where both inequalities are true simultaneously, marking the valid solution region. This method of shading different regions allows for easy visualization of where inequalities overlap. Graphical representation aids in understanding complex inequalities and simplifies identifying overlap areas, hence making interpreting solutions clearer.