Problem 18 - Algebra 1: Concepts and Skills

Question

Graph the system of linear inequalities. $$ \begin{array}{r} {x+y \leq 6} \\ {x \geq 1} \\ {y \geq 0} \end{array} $$

Step-by-Step Solution

Answer

The solution to the system of inequalities is the region that lies to the right of the line \(x = 1\), above the line \(y = 0\), and below the line \(y = -x + 6\).

Step 1: Draw the lines

Firstly, draw the lines corresponding to each of the three inequalities. These are \(y = -x + 6\), \(x = 1\), and \(y = 0\).

Step 2: Shade the regions

Secondly, shade the region where \(x+y \leq 6\), which is the region below the line; where \(x \geq 1\), which is the region to the right of the line; and where \(y \geq 0\), which is the region above the line.

Step 3: Identify the overlap

Finally, identify the region where all these three shaded areas overlap. This will be the solution to this system of inequalities.

Key Concepts

Graphing Inequalities

When it comes to graphing inequalities, it's helpful to begin with a basic understanding of graphing linear equations. For equations, we usually plot a line on the graph. However, inequalities don't have just one answer — they have a whole range of solutions! This is why we need to think in terms of regions rather than lines.

For example, when dealing with an inequality such as the first one in our exercise, \(x+y \leq 6\), we begin by graphing the line \(y = -x + 6\) as if it were an equation. This line divides the graph into two regions: above and below the line. To determine which region represents the inequality's solutions, we can use a test point. If the test point satisfies the inequality, then that entire region is the solution. In our exercise, the region below the line is shaded as it satisfies the inequality. This method of shading provides a clear visual of the potential solutions.

Linear Inequalities

Linear inequalities are very similar to linear equations but with one key difference: they use inequality symbols such as \(<\), \(>\), \(\leq\), or \(\geq\) instead of an equals sign. This means that the solutions to these inequalities are not just single points or lines, but entire areas on the graph.

For instance, the second inequality from our exercise, \(x \geq 1\), shows that we're interested in all the points where \(x\) is greater than or equal to 1. To represent this on a graph, we draw a vertical line at \(x = 1\) and then shade everything to the right of the line, because all of those points have x-values that meet the inequality's condition — they are all greater than or equal to 1.

Solution Region

Finding the solution region is like completing a puzzle. It's the area where all of the shaded regions from the inequalities intersect. This common area represents all the ordered pairs that satisfy all the inequalities in the system simultaneously.

In the exercise, once we've graphed and shaded our inequalities, the solution region is the area that is shaded for all three conditions: it is below the line \(y = -x + 6\), to the right of the line \(x = 1\), and above the line \(y = 0\). It's a bounded area where if you pick any point, that point will satisfy all of the given inequalities. Carefully graphing each inequality and then finding the intersection of these areas ensures that we've correctly found the solution region.

Inequalities in Two Variables

When working with inequalities in two variables, like the ones in our exercise (\(x+y \leq 6\), \(x \geq 1\), \(y \geq 0\)), each inequality gives us a half-plane of the coordinate system as its solution set. Half-planes are so named because they cover half of the coordinate plane. Our job is to identify which half-plane is the solution set for each inequality, and this is done by first considering the related equation - the boundary line.

The boundary line, which we draw with a solid or dashed line depending on whether the inequality includes equality (\(\leq\), \(\geq\)) or not (\(<\), \(>\)), helps us see where the 'cut-off' point is for each inequality. With a solid line, points on the line are included in the solution; with a dashed line, they are not. Every point on one side of this line will satisfy the inequality, and every point on the other side will not. Once we determine the correct side for each inequality, overlapping these half-planes tells us the overall solution set for the system.