Problem 29 - Algebra 2

Question

Solve each system of inequalities by graphing. \(\left\\{\begin{array}{c}{x-2 y \geq 11} \\ {5 x+4 y<27}\end{array}\right.\)

Step-by-Step Solution

Answer

The solution is the overlapped region that lies below the line \(y = x - 4\) and outside the V-shaped graph of \(y = |x - 6|\).

Step 1: Graph the first inequality

Graph the line of the equation \(y = x - 4\). Since the inequality is \(y \leq x - 4\), shade the area below this line because this is the area where the values of \(y\) are less than or equal to those on the line.

Step 2: Graph the second inequality

Graph the equation \(y = |x - 6|\) by plotting two lines: one for \(y = x - 6\) when \(x \geq 6\), and another for \(y = -(x - 6)\) when \(x < 6\). Since the inequality is \(y > |x - 6|\), shade the area outside and above these V-shaped lines, because this is where the values of \(y\) are greater than those on the lines.

Step 3: Find the region of intersection

Look for the region where the shading for both inequalities overlaps. This region is the solution to the system of inequalities.

Key Concepts

Graphing Inequalities

Understanding how to graph inequalities is a foundational skill in algebra. When graphing a linear inequality, like the one from our exercise, the first step is to graph the corresponding equation as if the inequality sign were an equals sign. This gives you the boundary line for the inequality. For the inequality \(y \<= x - 4\), we graph the line \(y = x - 4\).

Remember, the inequality sign tells us whether to shade above or below the line. If the inequality is \(y \<=\) or \(y \<\), we shade below the line since \(y\) is less than or equal to values on the line. Conversely, if it's \(y \geq\) or \(y >\), we shade above. Another tip is to use a dashed line for a strict inequality (where the points on the line aren't included, such as \(y \<\) or \(y >\)) and a solid line when the inequality includes the value on the line (\(y \leq\) or \(y \geq\)).

Absolute Value Inequalities

Absolute value inequalities involve expressions within absolute value bars, like \(|x - 6|\) in our example. The absolute value causes the expression to have two scenarios: one for the positive outcome and one for the negative outcome of the contents of the absolute value.

In the case of the inequality we're dealing with, \(y > |x - 6|\), we interpret it by creating two separate inequalities: \(y > x - 6\) for the scenario where \(x \geq 6\), and \(y > -(x - 6)\) for the scenario where \(x < 6\). This generates a V-shaped graph, where the region of interest is outside of the 'V' since \(y\) is greater than the absolute value expression. Typically, when graphing an absolute value inequality, you would use open dots to indicate that points on the 'V' lines themselves are not included in the solution due to the strict inequality \(y >\).

Shading Solution Regions

Once you've graphed your inequalities on the same coordinate plane, the solution to the system is where their shaded regions overlap. This region represents all the points that satisfy both inequalities at the same time. In our example exercise, we look for where the shading below the line of \(y \leq x - 4\) intersects with the shading outside the V-shaped graph of \(y > |x - 6|\).

The area where both conditions are true is the solution region. It's important to cross-check that every point in that region satisfies both inequalities of the system. If no such overlapping region exists, then the system has no solution. Students should get into the practice of clearly indicating the solution region, often by using a different shading pattern or marking it with points, to eliminate any confusion about where the solutions lie.