Explainer: One-step Inequalities: Addition or Subtraction

In this explainer, we will learn how to solve one-step linear inequalities by addition or subtraction.

An inequality is a mathematical sentence containing an inequality symbol (<,,>,or); it shows that the value of one expression is greater than the value of another, while an equation states the equality of two expressions. When a variable is involved in an inequality, solving an inequality means finding the range of values that the variable can take to make the inequality true. This range of values is given as an inequality in the form 𝑥(<,,>,or)𝑐, where 𝑐 is a number.

Often, we come across more complex inequalities, for instance, 𝑥+5>8. In this explainer, we are going to learn how to simplify this type of inequalities to get an inequality of the form 𝑥(<,,>,or)𝑐, where 𝑐 is a number.

Let us recall the balance method for equations. An equation states the equality between two expressions; we can represent it with a balance scale. For instance, the equation represented on the diagram is 𝑥+5=8.

Now, we do not want to solve the equation 𝑥+5=8 but the inequality 𝑥+5>8. Not only do we know that 𝑥+5 is not equal to 8 but also that 𝑥+5 is greater than 8. We can represent this on an unbalanced scale by considering that 𝑥+5 weighs more than 8, so that its side is lower than the side where there is a lighter weight (here 8).

Now, as with a real balance scale, if the same weight is removed from or added to each side of the scale, the side on the left will still be heavier than the side on the right.

We can write this formally as

that is, 𝑥>3.

We can summarize this by writing the addition and subtraction rule for inequalities.

Addition and Subtraction Rule for Inequalities

An inequality still holds true when the same number is added to or subtracted from both of its sides.

Let us look at some examples to see how to apply this rule to simplify inequalities.

Example 1: Finding an Equivalent Inequality by Using the Addition and Subtraction Rule

Complete using , <, >, or : If 𝑏<5, then 𝑏+14.

Answer

We start here with 𝑏<5 and we want to find something about 𝑏+1. We know that we can add the same number to both sides of an inequality and it will still hold true. So, we have

that is, 𝑏+1<4.

Hence, the inequality should be completed using <.

Example 2: Solving an Inequality by Adding

Given that 𝑥43>28, what number do you need to add both sides of the inequality to colve for 𝑥

Answer

To solve 𝑥43>28, we need to have 𝑥 on its own on one side. For this, we add 43 to both sides, so that we will have 𝑥43+43=𝑥 on the left-hand side, while we have 28+43=71 on the right-hand side. This will give 𝑥>71.

Hence, the answer is that we need to add 43 to both sides of the inequality.

Example 3: Solving an Inequality by Subtracting

If 𝑥+6>9, then 𝑥>.

  1. 3
  2. 54
  3. 15
  4. 3

Answer

We start here with the inequality 𝑥+6>9. To solve it for 𝑥, we need to have 𝑥 on its own on one side. For this, we take away 6 from each side:

that is, 𝑥>3.

Example 4: Solving an Inequality by Subtracting

If 𝑎+4538, then .

  1. 𝑎7
  2. 𝑎83
  3. 𝑎7
  4. 𝑎83
  5. 𝑎83

Answer

We start here with the inequality 𝑎+4538. To solve for 𝑎, we need to have 𝑎 on its own on one side. For this, we take away 45 from each side:

that is, 𝑎83.

Example 5: Finding an Equivalent Inequality Using the Addition and Subtraction Rule

Select the option that is equivalent to 𝑎+7<0.

  1. 𝑎>7
  2. 7𝑎
  3. 𝑎7
  4. 7>𝑎

Answer

We start here with the inequality 𝑎+7<0. To solve it for 𝑎, we need to have 𝑎 on its own on one side. For this, we take away 7 from each side:

that is, 𝑎<7.

We have solved our inequality; however, none of the options given in the questions displays what we have found. We need to inspect the options more closely to realize that one of them is equivalent to our solution; namely, 7>𝑎 (option D). Indeed, if we draw 𝑎 and 𝑏 on a number line so that 𝑎 is less than 𝑏, then we see that 𝑏 is greater than 𝑎.

So, 𝑎<𝑏 and 𝑏>𝑎 are equivalent.

Hence, our answer is 7>𝑎 (option D).

Key Points

  1. An inequality is a mathematical sentence containing an inequality symbol (<,,>,or); it shows that the value of one expression is greater than that of another.
  2. To rearrange an inequality of the form 𝑥+𝑎(<,,>,or)𝑏 to the form 𝑥(<,,>,or)𝑐, we use the addition and subtraction rule for inequalities.
  3. The addition and subtraction rule for inequalities states that an inequality still holds true when the same number is added to or subtracted from both of its sides.
  4. If the number 𝑎 is less than the number 𝑏, then 𝑏 is greater than 𝑎.
    This means that the inequality 𝑎<𝑏 is equivalent to 𝑏>𝑎. With more complex inequalities, we have, for instance, that 𝑥+12𝑦6 is equivalent to 𝑦6𝑥+12.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.