Question

Graph the solution set of the system of inequalities. $$\left\\{\begin{aligned} 3 x+2 y &<6 \\ x &>1 \\ y &>0 \end{aligned}\right.$$

Short answer

The solution set to this system of inequalities is the intersection of the regions corresponding to the three inequalities. This region is the area above the x-axis, to the right of the line \(x = 1\), and below the line defined by \(3x + 2y = 6\).

Step-by-step solution

Graph the first inequality

Begin with the inequality \(3x + 2y < 6\). The boundary line for this inequality is given by the equation \(3x + 2y = 6\), which can be easily graphed as a straight line. To decide which side of the boundary line to shade, a simple test point may be used such as the origin \((0,0)\) since it does not lie on the line itself. Evaluating \(3(0) + 2(0) < 6\) results in \(0 < 6\), which is true. Therefore, the region to be shaded for this inequality exists below the boundary line.

Graph the second inequality

Next, graph the inequality \(x > 1\). The boundary line for this inequality is the vertical line \(x = 1\). Here, the region to consider for shading would be all the points greater than x = 1. Hence, the region to the right of the line should be shaded.

Graph the third inequality

Lastly, consider the inequality \(y > 0\). The boundary line for this inequality is the horizontal line \(y = 0\). For this inequality, the region where all y values are positive (above) this line should be shaded.

Identify the intersection

The solution to the entire system is the intersection of the individual solution sets from the previous steps. This region in the intersection is where all shading overlaps. It is important to consider the system of inequalities as a whole to identify the feasible region correctly.

Key concepts

Linear Inequalities

Understanding linear inequalities is crucial when you're diving into the world of algebra. Similar to linear equations, these inequalities compare two expressions using inequality symbols like '>' for greater than, or '<' for less than. While a linear equation forms a line, a linear inequality divides the coordinate plane into two regions: one that satisfies the inequality, and one that doesn't.

For example, take the inequality \(3x + 2y < 6\). This inequality implies that all points \((x, y)\) that make the expression \(3x + 2y\) less than 6 are part of the solution set. When graphing, this will create a 'half-plane' — an entire side of the corresponding boundary line is included in the solution set, rather than just the line itself. It's helpful to think of this as a decision boundary separating points that pass the 'test' from those that don't.

Inequality Shading

The process of inequality shading is like coloring within the lines, but with a mathematical twist. Once you've graphed the boundary line, you must determine which side of it contains the solutions to your inequality. The 'shading' represents all the points that satisfy the inequality.

To decide where to 'shade':

• Pick a test point not on the boundary line.
• Substitute the values of this point into the inequality.
• If the inequality holds true, then the region containing that point is the solution region, and that's where you shade.

If you choose \((0,0)\) as your test point and the inequality holds true, as with \(3(0) + 2(0) < 6\), you'll shade the half-plane that contains the origin. If the origin does not satisfy the inequality, you'll shade the opposite side.

Feasible Solution Region

The feasible solution region sounds fancy, but it's just the sweet spot where all the conditions of a system of inequalities are met. In other words, find where the shaded areas of all the inequalities overlap, and that's your jackpot.

In a system of multiple inequalities, like the one with \(3x + 2y < 6\text{, } x > 1\text{, and } y > 0\), each creates its own shaded area on the graph. However, we're interested in where all these regions come together. This shared zone is the place where all the inequalities are true at the same time. To find it, simply look for the area on your graph that has been shaded over by each inequality. That's where the feasible solution lies – an area where all conditions are happy to coexist and any point within it is a solution to the system.

Boundary Lines

When we talk about boundary lines in graphing inequalities, we're referring to the lines that define the edges of our solution regions. They may look like your ordinary graphed lines, but they have a secret: they show you the threshold where the inequality changes from true to false.

These lines are graphed using the related linear equation, which is essentially the inequality stripped of its 'inequality' part — the '<' becomes '=', for example. Yet, not all boundary lines are created equal. If the original inequality includes the equal sign, like \(\leq\) or \(\geq\), the boundary line is solid, indicating that the points on the line are included in the solution set. If not, the line is dashed, warning that points on the line itself are not solutions. In our system, \(x > 1\) and \(y > 0\) would be graphed with dashed lines, signaling they're not part of the solution.