Linear+Systems-Target+E-Where+is+the+Overlap?-Guided+Learning

**Target E: Graph a System of a Linear Inequalities**

 * [[image:Discuss this.png width="173" height="97"]] || Heidi is working two part-time jobs to earn some extra money. She can't work more than 10 hours per week because of school responsibilities. If she earns $8 per hour baby sitting and $10 per hour tutoring, what are at least three possbile ways that she can make at least $84.00 in one week? ||

Graphing a system of inequalities is very similar to graphing a system of equations, but just a little different. A system of inequalities is just two or more inequalities that are graphed on the same coordinate axis where the solutions are solutions to both of the inequalites (or the overlap of the shaded regions). Let's take a look at the following system of inequalities. math y<\dfrac{1}{2}x-3\ math
 * Example 1:**

math 3x+4y\leq 12\ math The first picture represents the graph of the first inequality. The blue shaded region represents all of the solutions to the first inequality. The second picture represents the graph of the second inequality from the system. The red shaded region represents the solutions to the second inequality. The third picture represents the graph of the system of inequalities. As you can see, the graph is formed by placing the graphs of the two separate inequalities together on the same coordinate plane. The solutions to the overall system of inequalities are represented in the portion of the graph where the two shaded areas overlap (pink/purple area).

Application Problems
We can also solve application problems involving systems of linear inequalities. This falls under Writing Linear Functions-Target C but we put it in this section due to the fact that we just learned how to graph inequalities. Let's try the following problem.

You are planning a birthday party for your friend Grace. Everyone is bringing snacks but she loves loves to drink orange soda and eat hamburger sliders. You have at most $200 to spend on these two food items. If each soda costs $1.00 and sliders cost $2 each. You would like to have at least 60 sliders. Write a system of linear inequalities to represent the situation. Find a possible combination of sliders and orange sodas that you could buy for Grace's party.
 * Example 2**:

The first inequality that we will need to write will deal with each of the two items. If you recall writing this inequality is similar to the equations we wrote in Writing Linear Functions-Target C when we wrote an equation in standard form. You can revisit Writing Linear Functions - Target C - Guided Learning if you need to refresh your memory. The only difference is this is an inequality where we only have at most $200 to spend. The second inequality will be written base on the statement "you would like to have at least 60 sliders." This inequality is only based on sliders which means it will only involve one variable, y, which we are using to represent the number of sliders. The other piece of information is at least - which means you want 60 or more sliders. math y\geq 60\ math
 * We will let x represent the number of orange sodas that each cost $1 per soda
 * We will let y represent the number of sliders each costing $2 per burger
 * We have at most $200 to spend so instead of an equals sign we will use a less than or equal to sign because we can spend up to $200, which means we could spend less

We will now graph the system of inequalities and look for the area of overlap. For the first inequality, it may be best to use the x and y intercepts when you are graphing. For the second inequality, the boundary line is horizontal. math y\geq 60\ math

Now the graph. Make sure you label the axes and scale your axes appropriately. In the graph below, the red line represents our first inequality and the blue line represents our second inequality. The overlap region is shaded in pink. This is where you would find your solutions. Remember if you have questions about the graphing of the individual inequalities refer back to Writing Linear Functions - Target D Guided Learning. If you have trouble with finding the overlapping region, refer back to example 1.



We are trying to find the number of orange sodas and sliders you could buy, you need to stay within your budget of $200 and purchase at least 60 sliders. Any ordered pair in the pink shaded region will work, including the solutions that fall on both of the lines. Make sure to pick a solution that is a whole number. We can't buy a half of a slider or half of a soda. One possible choice would be 40 orange sodas and 70 sliders. **Writing a Linear Inequality from a Graph** In the next example, we will start with the graph and write the linear inequlity from it. We will use function form (slope intercept form) to do this. If you forget how to write an equation in function form, please refer back to Wrting Linear Functions - Target A Guided Learning.

As you recall, the steps to writing an equation in function form are below, what we will have to consider is which inequality sign we will used based on the type of boundary line, solid or dashed and the shading, above or below.
 * Example 3:** Write a linear inequality for the following graph.

1) Identify the slope: Using rise over run we can see that the slope value is 2.  2) Identify a point: We would use this if we did not have the y-intercept, but in this graph we do have it so go to step 3. 3) Find the y-intercept: (0, 4)  4) Determine the correct inequality sign to use (determined by type of line and shading): Since the region is shaded below the boundary line and dashed we will use a < sign.

Our linear inequality will be: y < 2x + 4. A good way to check to see if you wrote the linear inequality correctly would be to graph the linear inequality you just wrote and see if it matches with the orignal graph.

**Example 4**: Write a system of linear inequalities for the following graph.

If we look at the graph, you can see that there is more than one linear inequality. We will need to write two linear inequalities. We will follow the same process as in Example 4 to obtain our system of linear inequalities. It is important that you pay attention to which shading belongs to what line.

1) Graph the following system of linear inequalites.
 * Quick Check - Linear Systems - Target E - Where is the Overlap **

math y<\text{-}\dfrac{2}{3}x-2 math

math y\geq 2x-4 math

2) Karen's goal is to make at least $100 this week. She makes $10 per hour babysitting and $8 per hour working at the swim clubs snack shack. She can only work up to 6 hours at the snack shack. Write the system of linear inequalities to model the problem.

3) Write a system of linear inequalities based on the graph below.

Quick Check Solutions - Systems of Linear Inequalities - Target E - Where is the Overlap?