Systems of Inequalities
Word Problems

As you get further into Algebra 1, you will find that the real world problems become more complex. They have more questions to be answered and require more steps to find the solution.

When you get into systems of inequalities, this is especially true because you are dealing with two inequalities.

But... don't let that intimidate you! You have all the skills that you need to solve these problems. Take one step at a time and think about what you need in order to answer the question. Read through my example very carefully, and study how I performed each step. Pay careful attention to the key words (highlighted words) and how each inequality was written based on the problem.

Then complete the practice problems. You can do it!


Example 1 - Systems of Inequalities Word Problem

Sarah is selling bracelets and earrings to make money for summer vacation. The bracelets cost $2 and earrings cost $3. She needs to make at least $500.

  • Write an inequality to represent the income from the jewelry sold.
  • Sarah knows that she will see more than 50 bracelets. Write an inequality to represent this situation.
  • Graph the two inequalities and shade the intersection.
  • Identify a solution. How many bracelets and earrings can Sarah sell?

Solution

Step 1: Highlight the important information in the problem.

Sarah is selling bracelets and earrings to make money for summer vacation. The bracelets cost $2 and earrings cost $3. She needs to make at least $500.


Step 2: Identify your variables.  Think about what you don't know and need to know to solve the problem.

Let x = the number of bracelets sold.

Let y = the number of earrings sold.


Step 3: Write an inequality to represent the income form the jewelry sold.

2x + 3y > 500


Here's how I came up with this inequality.

Explanation of how to write an inequality

Step 4: Sarah knows that she will sell more than 50 bracelets. Write an inequality to represent this situation.

x > 50

Example of how to write an inequality

Step 5: Graph the two inequalities and shade the intersection.

How to graph a system of inequalities
Systems of inequalities graph
Justifying an answer to a system of inequalities word problem

For most real world problems, it will be easiest to graph the inequalities using the x and y intercepts. Make sure that you scale your grid so that both inequalities can be graphed on the same grid.

Here's a few tips to help you as you work through your word problems.


Steps for Solving a System of Inequalities Word Problem

  • Read the problem and highlight important information.

  • Identify the variables.

  • Find one piece of information in the problem that you can use to write an inequality.

  • Find a different piece of information that you can use to write a second inequality.

  • Graph both inequalities on a grid. Make sure you use appropriate boundary lines and shade the correct half plane for each inequality.

  • Identify the intersection of the two inequalities and answer the questions that pertain to the problem.

Now it's time to move onto the practice problems. You can do it!
Click here to move onto the systems practice problems.



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