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4. Equations & Inequalities
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United States of AmericaVirginia
Grade 7

4.07 Solve problems with inequalities

Lesson
Practice

Solve problems with inequalities

Much like solving equations from real world problems, there are certain keywords or phrases to look out for. When it comes to inequalities, we now have a few extra keywords and phrases to represent the different inequality symbols.

Keywords:

  • \gt greater than, more than.

  • \geq greater than or equal to, at least, no less than.

  • \lt less than.

  • \leq less than or equal to, at most, no more than.

When working to solve problems always be sure to check your answer to see if it is reasonable and then check it by substituting into your inequality.

Examples

Example 1

Consider the following situation:

"The sum of 2 groups of x and 1 is at least 7."

a

Construct and solve an inequality for the given situation.

Worked Solution
Create a strategy

Translate the phrases into mathematical symbols then solve the inequality by isolating the x on side of the inequality.

Apply the idea

The phrase "at least" means the same as "greater than or equal to", "groups of" means multiplication, and "sum" means addition.

\displaystyle 2x+1\displaystyle \geq\displaystyle 7Write the inequalty
\displaystyle 2x+1-1\displaystyle \geq\displaystyle 7-1Subtract 1 from both sides
\displaystyle 2x\displaystyle \geq\displaystyle 6Simplify
\displaystyle \dfrac{2x}{2}\displaystyle \geq\displaystyle \dfrac{6}{3}Divide both sides by 2
\displaystyle x\displaystyle \geq\displaystyle 3Simplify

Using a number line, the solution to this inequality is:

-2-1012345678910

So the possible values of x are those that are greater than or equal to 3.

b

Does the solution x=-3 satisfy the inequality?

Worked Solution
Create a strategy

Substitute the value of x into the inequality and evaluate.

Apply the idea
\displaystyle 2x+1\displaystyle \geq\displaystyle 7
\displaystyle 2\left(-3\right)+1\displaystyle \geq\displaystyle 7Substitution property
\displaystyle -6+1\displaystyle \geq\displaystyle 7Evaluate the multiplication
\displaystyle -5\displaystyle \geq\displaystyle 7Evaluate the addition

Because -5 \ngeq 7, x=-3 does not satisfy the inequality and is not a solution.

Example 2

Lachlan is planning on going on vacation. He has saved \$2118.40, and spends \$488.30 on his airplane ticket.

a

Let x represent the amount of money Lachlan spends on the rest of his holiday. Write an inequality to represent the situation, and then solve for x.

Worked Solution
Create a strategy

The amount of money that he can spend on his holiday is up to but not more than the difference between his savings and the amount spent on the airplane ticket. Translate this information into mathematical symbols.

Apply the idea

The phrase "up to but not more than" means that we are going to use the \leq symbol and the phrase "difference between" means subtraction.

\displaystyle x\displaystyle \leq\displaystyle 2118.40 - 488.30Write the inequality
\displaystyle x\displaystyle \leq\displaystyle \$1630.10Evaluate
b

What is the most that Lachlan could spend on the rest of his holiday?

Worked Solution
Create a strategy

Determine the largest value of the inequality from part (a) by recalling the definition of inequality symbol.

Apply the idea

The inequality x \leq \$1630.10 means that x can take any value that is less than or equal to \$1630.10. So the maximum amount that Lachlan can spend on his holiday is \$1630.10.

Example 3

At a sport clubhouse the coach wants to rope off a rectangular area that is adjacent to the building. He uses the length of the building as one side of the area, which measures 26 meters. He has at most 42 meters of rope available to use.

If the width of the roped area is W, form an inequality and solve for the range of possible widths.

Worked Solution
Create a strategy

Translate the given information into mathematical symbols and use the perimeter equation to solve for the possible width of the rope.

Apply the idea

One side of the roped area is the length of the clubhouse. So the situation looks something like this:

A layout of the clubhouse and the roped section. Ask your teacher for more information.

The length of the roped area is equal to the length of the clubhouse which is 26 meters.

\displaystyle \text{width + width + length} \displaystyle =\displaystyle \text{Perimeter}Rope perimeter is equal to the sum of two widths and the length
\displaystyle 2W+26\displaystyle \leq\displaystyle 42Substitute the values and variables
\displaystyle 2W + 26 - 26\displaystyle \leq\displaystyle 42 - 26Subtract 26 from both sides
\displaystyle 2W\displaystyle \leq\displaystyle 16Simplify
\displaystyle \dfrac{2W}{2}\displaystyle \leq\displaystyle \dfrac{16}{2}Divide both sides by 2
\displaystyle W\displaystyle \leq\displaystyle 8\text{ meters}Simplify

The maximum width should be 8 meters to be able to use all of the 42-meter rope and we will have enough.

Example 4

The student government association is planning a seventh grade dance as a fundraiser. It costs \$250 to pay for the DJ and decorations. Tickets to the dance cost \$8 each. How many tickets must they sell to raise at least \$1000?

Worked Solution
Create a strategy

Translate the given information into an inequality and use that to determine how many tickets they must sell.

Apply the idea

Each ticket costs \$8, so that will be the value that is multiplied by the variable t. The cost of the DJ and decorations must be subtracted from the amount they raise. They need to raise at least \$1000, so their amount must be greater than or equal to \$1000, so we will use the symbol \geq.

We can use the inequality 8t-250 \geq 1000 to find how many tickets they must sell.

\displaystyle 8t-250\displaystyle \geq\displaystyle 1000
\displaystyle 8t-250+250\displaystyle \geq\displaystyle 1000+250Addition property of inequality
\displaystyle 8t\displaystyle \geq\displaystyle 1250Simplify
\displaystyle \frac{8t}{8}\displaystyle \geq\displaystyle \frac{1250}{8}Division property of inequality
\displaystyle t\displaystyle \geq\displaystyle 156.25Simplify

The student government association must sell at least 156.25 tickets. Since tickets must be sold in whole amounts, they must sell 157 tickets or more.

Example 5

Consider the following inequality: 3.7+ 8 x \geq 19.5

a

Create a real world scenario that could be represented by the equation.

Worked Solution
Create a strategy

We know that x represents a quantity that will occur more than once and 3.7 only occurs once. The total of those amounts must be greater than or equal to 19.5.

Apply the idea

Here is one possible scenario:

Luella needs to log at least 19.5 hours in her reading log. So far, she has logged 3.7 hours. She has eight days left to reach her goal. This inequality will help determine the amount of hours Luella needs to read in the reamining time to reach her goal. x represents the number of hours spent reading each day.

b

Solve the inequality and explain the answer in context to the scenario you created.

Worked Solution
Create a strategy

Use the properties of inequality to solve the inequality.

Apply the idea
\displaystyle 3.7+ 8x\displaystyle \geq\displaystyle 19.5
\displaystyle 3.7-3.7+8x\displaystyle \geq\displaystyle 19.5-3.7Subtraction property of inequality
\displaystyle 8x\displaystyle \geq\displaystyle 15.8Evaluate the subtraction
\displaystyle \dfrac{8x}{8}\displaystyle \geq\displaystyle \dfrac{15.8}{8}Division property of inequality
\displaystyle x\displaystyle \geq\displaystyle 1.975Evaluate the division

Luella needs to read at least 1.975 hours per day to reach her goal.

Reflect and check

We can check our solution by substituting the value we found for x back into the inequality.

\displaystyle 3.7+8x\displaystyle \geq\displaystyle 19.5
\displaystyle 3.7+8\left(1.975\right)\displaystyle \geq\displaystyle 19.5Substitution property
\displaystyle 3.7+15.8\displaystyle \geq\displaystyle 19.5Evaluate the multiplication
\displaystyle 19.5\displaystyle \geq\displaystyle 19.5Evaluate the addition
Idea summary

Keywords and phrases to represent the different inequality symbols:

  • \gt greater than, more than.

  • \geq greater than or equal to, at least, no less than.

  • \lt less than.

  • \leq less than or equal to, at most, no more than.

Outcomes

7.PFA.4

The student will write and solve one- and two-step linear inequalities in one variable, including problems in context, that require the solution of a one- and two-step linear inequality in one variable.

7.PFA.4d

Write one- or two-step linear inequalities in one variable to represent a verbal situation, including those in context.

7.PFA.4e

Create a verbal situation in context given a one or two-step linear inequality in one variable.

7.PFA.4f

Solve problems in context that require the solution of a one- or two-step inequality.

7.PFA.4g

Identify a numerical value(s) that is part of the solution set of as given one- or two-step linear inequality in one variable.

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