Topics
6. Equations & Inequalities
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United States of America
Grade 7

6.06 One-step inequalities

Lesson
Practice

Operations with inequalities

When we manipulate equations, we can apply the same operation to both sides and the equation remains true. Consider the following equation:

\displaystyle x + 7\displaystyle =\displaystyle 12
\displaystyle x+7-7\displaystyle =\displaystyle 12-7Subtract 7 from both sides
\displaystyle x\displaystyle =\displaystyle 5Evaluate

When working with inequalities, this is not necessarily always the case.

Consider the inequality 9<15.

Let's add 3 to 9 and 15:

\displaystyle 9+3\displaystyle <\displaystyle 15+3
\displaystyle 12\displaystyle <\displaystyle 18

Now try subtracting 3 from 9 and 15:

\displaystyle 9-3\displaystyle <\displaystyle 15-3
\displaystyle 6\displaystyle <\displaystyle 12

If we add or subtract both sides by any number, we can see that the resulting inequality remains true.

If we added a negative that would be the same as subtracting, so we can also add and subtract negative numbers without changing the inequality.

What happens if we multiply or divide both sides?

Let's multiply by a positive:

\displaystyle 9\displaystyle <\displaystyle 15
\displaystyle 9\times3\displaystyle <\displaystyle 15\times 3
\displaystyle 27\displaystyle <\displaystyle 45The inequality symbol stays the same

Now try dividing by a positive:

\displaystyle 9\displaystyle <\displaystyle 15
\displaystyle \dfrac{9}{3}\displaystyle <\displaystyle \dfrac{15}{3}
\displaystyle 3\displaystyle <\displaystyle 5The inequality symbol stays the same

How about multiplying by a negative:

\displaystyle 9\displaystyle <\displaystyle 15
\displaystyle 9\times(-3)\displaystyle >\displaystyle 15\times(-3)
\displaystyle -27\displaystyle >\displaystyle -45The inequality switches from < \text{ to } >

And finally dividing by a negative:

\displaystyle 9\displaystyle <\displaystyle 15
\displaystyle \dfrac{9}{-3}\displaystyle <\displaystyle \dfrac{15}{-3}
\displaystyle -3\displaystyle >\displaystyle -5The inequality switches from < \text{ to } >

Examples

Example 1

Consider the following statement: 7<10

a

Add 6 to both sides of the inequality and simplify.

Worked Solution
Create a strategy

Perform the operation on both sides.

Apply the idea
\displaystyle 7+6\displaystyle <\displaystyle 10+6Add 6 from both sides
\displaystyle 13\displaystyle <\displaystyle 16Evaluate
b

After adding 6 to both sides, does the inequality still hold true?

Worked Solution
Create a strategy

Check if the resulting inequality on part (a) is still true.

Apply the idea

13 is less than 16, so the resulting inequality is still true.

Example 2

Consider the following statement: 6<10

a

Multiply both sides of the inequality by -4 and simplify. Do not change the sign of the inequality.

Worked Solution
Create a strategy

Perform the operation on both sides.

Apply the idea
\displaystyle 6 \times (-4)\displaystyle <\displaystyle 10 \times (-4)Multiply both sides by -4
\displaystyle -24\displaystyle <\displaystyle -40Evaluate
b

After multiplying both sides by -4, does the inequality still hold true?

Worked Solution
Create a strategy

Plot the numbers on a number line to determine if the resulting inequality on part (a) is still true.

Apply the idea

Plot the numbers -40 and -24 on the number line.

-48-44-40-36-32-28-24-20-16-12-8-40

Since -40 is farther to the left than -24, then it means it is less than -24. So the inequality is no longer true.

Idea summary

The following operations don't change the inequality symbol used:

  • Adding a number to both sides of an inequality.
  • Subtracting a number from both sides of an inequality.
  • Multiplying both sides of an inequality by a positive number.
  • Dividing both sides of an inequality by a positive number.

The following operations reverse the inequality symbol used:

  • Multiplying both sides of an inequality by a negative number.
  • Dividing both sides of an inequality by a negative number.

Solve one-step inequalities

Now that we have seen what happens when we perform operations with positive and negative numbers we can use this knowledge to solve inequalities.

Before jumping in algebraically, it can be helpful to consider some possible solutions and non-solutions. Then we can look at an algebraic strategy.

Examples

Example 3

Determine whether x=-2 satisfies the inequality x+3<4.

Worked Solution
Create a strategy

Substitute the value of x into the inequality, the value on the left hand side should be less than 4.

Apply the idea
\displaystyle x+3\displaystyle <\displaystyle 4
\displaystyle -2+3\displaystyle <\displaystyle 4Substitute the value of x
\displaystyle 1\displaystyle <\displaystyle 4Evaluate

x=-2 satisfies the inequality.

Reflect and check

Substituting the possible solution back into the inequality to see if the inequality remains true is a helpful strategy for checking your work.

Example 4

Solve -\dfrac{x}{2} \geq 5 for x.

Worked Solution
Create a strategy

We solve an inequality by isolating x on one side of the inequality.

Apply the idea
\displaystyle -\dfrac{x}{2}\displaystyle \geq\displaystyle 5
\displaystyle -\dfrac{x}{2} \times (-2)\displaystyle \leq\displaystyle 5 \times (-2)Multiply both sides by -2, reverse inequality symbol
\displaystyle x\displaystyle \leq\displaystyle -10Evaluate
Idea summary

To solve an inequality it can be helpful to consider some possible solutions and non-solutions by assigning values to the variable.

When solving an inequality:

  • Multiplying or dividing both sides by a negative number will reverse the inequality symbol.
  • Reversing the order of the inequality will reverse the inequality symbol too.

Graph solutions to one-step inequalities

As we have previously seen, we can  plot inequalities  on number lines.

We can first solve an inequality and then graph it. This can also help to check our answer.

When plotting an inequality:

  • The symbols < and > don't include the end point, which we show with an unfilled circle.
  • The symbols \geq and \leq do include the endpoint, which we show with a filled circle.

Examples

Example 5

Consider the inequality 1+x<2.

a

Solve the inequality for x.

Worked Solution
Create a strategy

We solve an inequality by isolating x on one side of the inequality.

Apply the idea
\displaystyle 1 +x\displaystyle <\displaystyle 2
\displaystyle 1-1+x\displaystyle <\displaystyle 2-1Subtract 1 from both sides
\displaystyle x\displaystyle <\displaystyle 1Evaluate
b

Now plot the solutions to the inequality 1+x<2 on the number line below.

-4-3-2-101234
Worked Solution
Create a strategy

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

Apply the idea

The inequality x<1 means that x can have any value less than but not equal to 1.

To show that 1 is not part of the solution, we will plot the point at 1 with a hollow circle. To show all values that are less than 1, we draw a ray from 1 pointing to the left.

-4-3-2-1012

Example 6

Consider the inequality 2x>-4.

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 2x\displaystyle >\displaystyle -4
\displaystyle \dfrac{2x}{2}\displaystyle >\displaystyle \dfrac{-4}{2}Divide both sides by 2
\displaystyle x\displaystyle >\displaystyle -2Evaluate
b

Now plot the solutions to the inequality 2x>-4 on the number line below.

-4-3-2-101234
Worked Solution
Create a strategy

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

Apply the idea

The inequality x>-2 means that x can have any value greater than but not equal to -2.

To show that -2 is not part of the solution, we will plot the point at -2 with an unfilled circle. To show all values that are greater than -2, we draw a ray from -2 pointing to the right.

-4-3-2-101234
Idea summary

When plotting an inequality:

  • The symbols < and > don't include the end point, which we show with an unfilled circle.
  • The symbols \geq and \leq do include the endpoint, which we show with a filled circle.

Outcomes

7.EE.B.4

Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.

7.EE.B.4.B

Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem.

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