Topics
4. Equations & Inequalities
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United States of AmericaVirginia
Grade 7

4.06 Solve two-step inequalities

Lesson
Practice

Solve two-step inequalities

Let's look at the inequality -3x+2 \geq 14 and think about the order of operations.

We can see that the operations applied to the variable x are:

  • First multiply by -3

  • Next add 2

To solve this inequality, we want to undo these operations in reverse order. So we will:

  • First, subtract 2 from both sides

  • Next, divide both sides by -3 (and reverse the inequality symbol)

All together this looks like:

\displaystyle -3x+2\displaystyle \geq\displaystyle 14Given inequality
\displaystyle -3x+2-2\displaystyle \geq\displaystyle 14-2Subtraction 2 from both sides
\displaystyle -3x\displaystyle \geq\displaystyle 12Evaluate the subtraction
\displaystyle \dfrac{-3x}{-3}\displaystyle \leq\displaystyle \dfrac{12}{-3}Divide both sides by -3 (reverse the inequality sign)
\displaystyle x\displaystyle \leq\displaystyle -4Evaluate the division

We found that x\leq-4. We can test some values in the given inequality to see if they result in a true statement. Let's try the numbers just above and below -4, say x=-5 and x=-3.

  • When x=-5, we have -3x+2 = -3 \cdot \left(-5\right) +2 = 17, which is greater than or equal to 14.

  • When x=-3, we have -3x+2 = -3 \cdot \left(-3\right) +2 = 11, which is not greater than or equal to 14.

So our result of x\leq-4 seems to be correct. We can also graph this on the number line. For x\leq-4, we will include -4 on the number line as a filled circle and a ray pointing to the left side.

-8-7-6-5-4-3-2-101234

Let's compare how we solved that inequality to how we solve a similar equation, -3x+2=14.

\displaystyle -3x+2\displaystyle =\displaystyle 14Given equation
\displaystyle -3x+2-2\displaystyle =\displaystyle 14-2Subtraction 2 from both sides
\displaystyle -3x\displaystyle =\displaystyle 12Evaluate the subtraction
\displaystyle \dfrac{-3x}{-3}\displaystyle =\displaystyle \dfrac{12}{-3}Divide both sides by -3
\displaystyle x\displaystyle =\displaystyle -4Evaluate the division

We can check the solution to the equation using the substitution property.

\displaystyle -3\left(-4\right)+2\displaystyle =\displaystyle 14Substitution property
\displaystyle 12+2\displaystyle =\displaystyle 14Evaluate the multiplication
\displaystyle 14\displaystyle =\displaystyle 14Evaluate the addition

Because 14=14, our solution is correct. We can graph the solution to the equation as a single point on a number line like this:

-8-7-6-5-4-3-2-101234

This graph differs from the graph of the solution to -3x+2\geq 14 because the solution to the equation 3x+2=14 is only one value while the solution set for the inequality contains an infinite number of values.

Examples

Example 1

Given 3x+27\gt 3

a

Solve the inequality.

Worked Solution
Create a strategy

Solve the inequality by isolating x on one side of the inequality.

Apply the idea
\displaystyle 3x+27-27\displaystyle >\displaystyle 3 -27Subtraction property of inequality
\displaystyle 3x\displaystyle >\displaystyle -24Evaluate the subtraction
\displaystyle \dfrac{3x}{3}\displaystyle >\displaystyle \dfrac{-24}{3}Division property of inequality
\displaystyle x\displaystyle >\displaystyle -8Evaluate the division
b

Determine if -8 is a solution to the inequality.

Worked Solution
Create a strategy

Determine if -8 is included in the possible values of x using the answer from part a.

Apply the idea

We know that x is any value that is greater than but not equal -8 when we solved the inequality in part a.

So, -8 is not solution to the inequality.

Reflect and check

We can also substitute -8 into the given inequality.

\displaystyle 3\left(-8\right)+27-27\displaystyle >\displaystyle 3-27Substitute x=-8
\displaystyle -24\displaystyle >\displaystyle -24Evaluate

-24 is not greater than -24, so -8 is not a solution to the inequality.

Example 2

Solve the following inequality: 4.2 \lt \dfrac{a}{5.2} + 3.1

Worked Solution
Create a strategy

Solve the inequality by isolating a on one side of the inequality.

Apply the idea
\displaystyle 4.2 -3.1 \displaystyle <\displaystyle \dfrac{a}{5.2} + 3.1 -3.1Subtraction property of inequality
\displaystyle 1.1\displaystyle <\displaystyle \dfrac{a}{5.2}Evaluate the subtraction
\displaystyle 1.1 \cdot 5.2\displaystyle <\displaystyle \dfrac{a}{5.2} \cdot 5.2Multiplication property of inequality
\displaystyle 5.72\displaystyle <\displaystyle aEvaluate the multiplication
Reflect and check

If what was given were an equation instead, we would have solved it similarly:

\displaystyle 4.2 -3.1 \displaystyle =\displaystyle \dfrac{a}{5.2} + 3.1 -3.1Subtraction property of equality
\displaystyle 1.1\displaystyle =\displaystyle \dfrac{a}{5.2}Evaluate the subtraction
\displaystyle 1.1 \cdot 5.2\displaystyle =\displaystyle \dfrac{a}{5.2} \cdot 5.2Multiplication property of equality
\displaystyle 5.72\displaystyle =\displaystyle aEvaluate the multiplication

Then we can just flip the equation to have the variable on the left side: a=5.72

However, if we do this with the given inequality, we need to reverse the inequality sign because we reversed the order of the entire inequality. So, using the symmetric property of inequality we can rewrite the solution as a \gt 5.72

Example 3

Consider the inequality -7-x \gt 13.

a

Solve the inequality.

Worked Solution
Create a strategy

Solve the inequality by isolating x on one side of the inequality.

Apply the idea
\displaystyle -7-x +7\displaystyle >\displaystyle 13+7Addition property of inequality
\displaystyle -x\displaystyle >\displaystyle 20Evaluate the addition
\displaystyle -1x\displaystyle >\displaystyle 20Multiplicative identity
\displaystyle \dfrac{-1x}{-1}\displaystyle <\displaystyle \dfrac{20}{-1}Division property of inequality, reverse the inequality sign
\displaystyle x\displaystyle <\displaystyle -20Evaluate the division
Reflect and check

If we had forgotten to reverse the inequality sign, we would have been left with an inequality that was not true. We would have gotten a solution of x \gt -20, let's test a point that satisfies the inequality to see if it is true. We can use x=4 to test.

\displaystyle -7–\left(4\right)\displaystyle >\displaystyle 13Substitution property
\displaystyle -11\displaystyle >\displaystyle 13Simplify

-11 \ngtr 13, so the inequality is false.

b

Now, plot the solutions to the inequality -7-x \gt 13 on a number line.

Worked Solution
Create a strategy

Plot the inequality from part (a) on the number line.

Apply the idea

The inequality x \lt -20 means that x can have any value less than but not equal to -20.

To show that -20 is not part of the solution, we will plot the point at -20 with an unfilled circle. To show all values that are less than -20, we draw a ray from -20 pointing to the left.

-25-24-23-22-21-20-19-18-17-16-15
Reflect and check

We can verify that we plotted the solution to the inequality correctly by selecting a value from the portion of the number line covered by the ray.

Let's use -22 as an example.

\displaystyle -7-\left(-22\right)\displaystyle >\displaystyle 13Substitute x=-22
\displaystyle 15\displaystyle >\displaystyle 13Evaluate

15 is indeed greater than 13, so we correctly plotted the solution on the number line.

Let's also use -20 to verify that its not part of the solution.

\displaystyle -7-\left(-20\right)\displaystyle >\displaystyle 13Substitute x=-20
\displaystyle 13\displaystyle >\displaystyle 13Evaluate

13 is not greater but equal to 13, so we are correct to exclude -20 when plotting the solution by using an unfilled circle.

Example 4

In an ecological park, you were assigned to manage the number of people who can go through the underground cave. According to safety guidelines, no more than 10 people can be in the cave at a time. Write an inequality to represent the number of people that can go through the underground cave at a time, given that there are 2 accompanying guides for each group going.

Worked Solution
Create a strategy

Let p represent the number of people. We should set the inequality to show that total number of people \left(p\right) going through the underground cave cannot be more than 10, including the accompanying guides.

Apply the idea

The cave can have 10 total people or less than that so wee need to use \leq.

p+2 \leq 10

Idea summary

When solving any inequality:

  • Multiplying or dividing both sides of an inequality by a negative number reverses the inequality symbol.

  • Writing an inequality in reverse order also reverses the inequality symbol.

When solving an inequality with two (or more) operations:

  • It is generally easiest to undo one operation at a time, in reverse order to the order of operations.

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.4a

Apply properties of real numbers and the addition, subtraction, multiplication, and division properties of inequality to solve one- and two-step inequalities in one variable. Coefficients and numeric terms will be rational.

7.PFA.4b

Investigate and explain how the solution set of a linear inequality is affected by multiplying or dividing both sides of the inequality statement by a rational number less than zero.

7.PFA.4c

Represent solutions to one- or two-step linear inequalities in one variable algebraically and graphically using a number line.

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.

7.PFA.4h

Describe the differences and similarities between solving linear inequalities in one variable and linear equations in one variable.

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