Topics
6. Equations & Inequalities
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United States of America
Grade 6

6.09 Problem solving with equations and inequalities

Lesson
Practice

Introduction

Previously, we represented numbers and wrote equations and inequalities. We also graphed and solved equations and inequalities. We are now ready to solve real world problems using the skills we have learned about equations and inequalities.

Problem solving with equations and inequalities

We may not notice it but equations and inequalities are very common in everyday life. From finding out if we have enough time, fuel or money, to speed limits and age restrictions, these not only can represent an exact amount, like in an equation, but can also represent a limit of what is allowed or what is possible, in an inequality.

Let's look at some strategies for solving real-world problems involving equations and inequalities:

  1. Read carefully to understand the entire problem.

  2. Identify what we are looking for by analyzing the question. Highlight the important information and keywords that we need to solve the problem.

  3. Identify and assign variables to the unknown quantities, for example, x or y.

  4. Translate the words by rewriting the given information in terms of the variables as equations or inequalities.

  5. Solve.

  6. Write your answer in a complete number sentence.

  7. Check if your answer is reasonable. Can you explain how you know?

Examples

Example 1

Ivan and Ned do some fundraising for their sporting team. Together, they raised \$371. If Ivan raised \$ m, and Ned raised \$157:

a

Write an equation that represents the relationship between the amounts each contributed.

Worked Solution
Create a strategy

Consider the relationship between the different amounts. Replace the words with mathematical symbols to form an equation.

Apply the idea

The keyword "together" tells us that we have to combine quantities.

Based on the problem, Ivan raised \$m while Ned raised \$157.

"Together" means the the sum of the two quantities which is equal to \$371.

We can write the equation as:

m+157=371

b

How much did Ivan raise?

Worked Solution
Create a strategy

Isolate the variable and simplify the equation to find m.

Apply the idea
\displaystyle m+157\displaystyle =\displaystyle 371We need to get m by itself.
\displaystyle m+157-157\displaystyle =\displaystyle 371-157Subtract 157 from both sides of the equation.
\displaystyle m\displaystyle =\displaystyle 214Simplify the equation.

Ivan raised \$214.

Reflect and check

Substituting the value of m which is 214, check to see if the equation m+257 is true.

In this case: 214+257=371 is true.

Example 2

To get a grade of C, Alvin must obtain a total score of at least 170 over two exams. In the first exam he achieved a score of 80. Let x represent what he must score on the last exam to get a C or better.

Write an inequality and solve for x.

Worked Solution
Create a strategy

Analyze the problem and look for keywords that we need to solve the problem.

Apply the idea

The "total" of the scores Alvin gets on his two exams needs to be "at least" 170.

Which operation do we use to find a "total"? Which inequality symbol is the same as "at least"?

We can find the total of the two scores by adding them together.

This needs to be greater than or equal to 170.

\displaystyle 80+x\displaystyle \geq\displaystyle 170Translate into an inequality.
\displaystyle 80+x-80\displaystyle \geq\displaystyle 170-80Subtract 80 from both sides of the equation to isolate x
\displaystyle x\displaystyle \geq\displaystyle 90Simplify

Alvin should get a score greater than or equal to 90.

Reflect and check

Is 80+90\geq 170?

What scores are greater than or equal to 90?

Idea summary

The following steps will help us solve real world problems involving equations and inequalities:

  1. Read carefully to understand the entire problem.

  2. Identify what we are looking for by analyzing the question. Highlight the important information and keywords that we need to solve the problem.

  3. Identify and assign variables to the unknown quantities, for example, x or y.

  4. Translate the words by rewriting the given information in terms of the variables as equations or inequalities.

  5. Solve.

  6. Write your answer in a complete number sentence.

  7. Check if your answer is reasonable. Can you explain how you know?

Outcomes

6.EE.B.7

Solve real-world and mathematical problems by writing and solving equations of the form xp=q and px=q for cases in which p, q and x are all nonnegative rational numbers.

6.EE.B.8

Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.

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