Question

Solve the following system graphically. \(y-x>-3\) \(y-2 x<2\) \(x+y-3<0\)

Short answer

Rewrite the inequalities in slope-intercept form: \(y > x - 3\), \(y < 2x + 2\), and \(y < -x + 3\). Plot the lines of equalities and determine the half-planes using test points. The solution region is the overlapping area of the shaded regions where all inequalities hold true. Points on the boundary lines are not included in the solution region.

Step-by-step solution

Rewrite inequalities in slope-intercept form

We will rewrite the inequalities in the slope-intercept form to easily plot them. 1. \(y - x > -3\) Adding x to both sides, we get: \(y > x - 3\) 2. \(y - 2x < 2\) Adding 2x to both sides, we get: \(y < 2x + 2\) 3. \(x + y - 3<0\) Subtracting x from both sides and adding 3, we get: \(y< -x + 3\)

Plot the lines of equalities

Now, we'll plot the lines of equalities on the Cartesian plane. These lines are: 1. \(y = x - 3\) (with slope 1 and y-intercept -3) 2. \(y = 2x + 2\) (with slope 2 and y-intercept 2) 3. \(y = -x + 3\) (with slope -1 and y-intercept 3)

Determine the half-planes

To determine the half-planes associated with each inequality, we'll use test points. For \(y > x - 3\), we can choose the test point (0,0): 0 > 0 - 3 (False) So, we shade the region below this line. For \(y < 2x + 2\), we can choose the test point (0,0): 0 < 2(0) + 2 (True) So, we shade the region below this line. For \(y < -x + 3\), we can choose the test point (0,0): 0 < - (0) + 3 (True) So, we shade the region below this line.

Identify the solution region

Finally, we identify the area where the shaded regions intersect. This area is the overlapping region, which is the solution to the given system. Note that the points on the boundary lines are not included in the solution region since the inequalities are "greater/less than" and not "greater/less than or equal to".

Key concepts

Slope-Intercept Form

Understanding the slope-intercept form is crucial when dealing with linear inequalities. It is expressed as \( y = mx + b \) , where m represents the slope of the line, and b indicates the y-intercept, or where the line intersects the y-axis.

To convert an inequality into slope-intercept form, manipulate the equation to isolate y on one side. This is what we do in our first step. By converting \( y - x > -3 \) into \( y > x - 3 \) using basic algebraic operations like addition or subtraction, we expose its slope (m=1) and y-intercept (b=-3). This form makes it much simpler to plot the inequality on a graph and understand its implications for the values of y relative to x.

Plotting Linear Inequalities

Plotting linear inequalities on a graph involves visualizing each inequality as a line and then determining on which side of that line the inequality holds true. The first step in plotting is to draw the lines corresponding to the equalities that match our inequalities.

For instance, to plot \( y = x - 3 \) you'd draw a straight line with a slope of 1 that crosses the y-axis at -3. However, because the original inequality is \( y > x - 3 \) we treat this line as a boundary. The actual set of solutions to the inequality lies on one side of this boundary line. The same process applies to the other inequalities; each line serves as a visual demarcation between areas on the graph that do and do not satisfy the inequality.

Shading Half-Planes

Once the boundary lines are in place, we use shading to indicate which side of each line is the solution to its respective inequality. This introduces us to the concept of shading half-planes. Shading is a crucial step, as it helps visually separate solutions from non-solutions on the graph.

A common technique to determine which side to shade is to choose a test point not on the line and substitute it into the inequality; the origin (0,0) is a handy choice if it’s not on the line itself. If the test point satisfies the inequality, that side is where all possible solutions can exist, and we shade it. For example, since (0,0) does not satisfy \( y > x - 3 \) the region above the line is shaded. This process is repeated for each inequality, with either the region above or below (or to the left/right) the boundary line being shaded.

Determining Solution Region

The solution region for a system of linear inequalities is where the shaded areas overlap. This overlapping region represents all the coordinate pairs that satisfy all inequalities simultaneously. It’s important to check the type of inequalities; if they are 'strict' (denoted by < or >), the boundary lines themselves are not included in the solution set.

In our example, after shading the correct half-planes for each inequality, we must locate the region where all shaded areas intersect. This is the graphical representation of the set of all solutions to the given system - a visual intersection of conditions defined by linear inequalities, helping us to apprehend the scope of possible solutions within the confines of the constraints laid out by the problem.