Explainer: Inequalities on a Number Line

In this explainer, we will learn how to find and deduce the one-variable linear inequality from a specific given interval represented on the number line.

An inequality is a mathematical sentence containing an inequality symbol (<,≀,>,β‰₯)or; it compares two numbers or expressions, saying which one is greater or less than the other. 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.

The range of values can be represented on a number line. Let us look at a couple of examples to see how.

Example 1: Representing the Set of Solutions to an Inequality on a Number Line

Which part of the shown line represents the set of solutions to the inequality π‘₯<27?

Answer

The given number line has been split into two regions. The blue region, region A, represents all numbers that are less than 27, and the pink region, region B, represents all numbers that are greater than 27.

Therefore, the inequality π‘₯<27 is represented by region A.

Example 2: Representing the Set of Solutions to an Inequality on a Number Line

Which part of the shown line represents the set of solutions to the inequality π‘₯>5?

Answer

The given number line has been split in two regions. The blue region, region A, represents all numbers that are less than 5, and the pink one, region B, represents all numbers that are greater than 5.

Therefore, the inequality π‘₯>5 is represented by region B.

With these two examples, we have seen that the set of solutions to an inequality of the form π‘₯<𝑐 or π‘₯>𝑐 is either all numbers less than 𝑐 for the first inequality or greater than 𝑐 for the second inequality.

Now, there is also the possibility that the variable π‘₯ be equal to the number 𝑐. We can represent this on a number line by using either a closed or an open circle at 𝑐: a closed circle means that the number 𝑐 does belong to the set of solutions, while an open circle means that 𝑐 does not belong to the set of solutions for π‘₯.

For instance, the inequality π‘₯≀0 has for solution all the numbers less than 0, including 0, since the inequality says that π‘₯ is less than or equal to 0. Therefore, the line representing the set of solutions contains a closed circle at 0.

And the inequality π‘₯>βˆ’8 has for solution all the numbers greater than βˆ’8, excluding βˆ’8, since the inequality says that π‘₯ is greater than βˆ’8. Therefore, the line representing the set of solutions contains an open circle at βˆ’8.

Let us look at an example of the solutions to an inequality represented in such a way.

Example 3: Identifying an Inequality as Represented on a Number Line

What inequality is represented on the given figure?

Answer

We see here that the blue line, representing the set of solutions to an inequality, contains all numbers greater than 4. Also, at 4, the line contains an open circle, meaning that the value of 4 is not included in the set of solutions. Therefore, we need to use the symbol β€œ>” for β€œgreater than,” and, therefore, the inequality represented on the figure is π‘₯>4.

Let us look now at inequalities in the form π‘Ž<π‘₯<𝑏 with π‘Ž<𝑏. They are called double inequalities because this type of inequalities contains two inequality symbols; it is actually two inequalities in one. It is π‘Ž<π‘₯π‘₯<𝑏and at the same time.

We can observe that if π‘Ž<π‘₯, then π‘₯>π‘Ž; both inequalities are indeed strictly equivalent. Note how one is obtained by reading the other from right to left! Therefore, we have π‘₯>π‘Žπ‘₯<𝑏.and

Let us look at the next example to see how such inequalities are represented on a number line.

Example 4: Representing an Inequality on a Number Line

Which part of the shown line represents the set of solutions to the inequality 3<π‘₯<56?

Answer

The inequality 3<π‘₯<56 is two inequalities in one. It means that we have at the same time π‘₯>3π‘₯<56.and

The set of solutions to π‘₯>3 is all numbers greater than 3, so it is represented by regions B and C.

The set of solutions to π‘₯<56 is all numbers less than 56, so it is represented by regions A and B.

For both inequalities to be true, then π‘₯ has to be in the region that is common to both sets, that is region B. We see that any number in region B is indeed greater than 3 and less than 56.

Let us now find an inequality from its set of solutions represented on a number line.

Example 5: Finding a Double Inequality from the Representation of Its Set of Solutions on a Number Line

What inequality is represented on the given figure?

Answer

The region shown in the diagrams represents all numbers greater than βˆ’4 and less than 2. The line contains an open circle at both βˆ’4 and 2, meaning that these values are not part of the solution, so we have at the same time π‘₯>βˆ’4π‘₯<2.and

This can be written as the double inequality βˆ’4<π‘₯<2.

Key Points

  1. An inequality is a mathematical sentence containing an inequality symbol(<,≀,>,β‰₯)or
  2. Solving an inequality where a variable is involved means finding the range of values that the variable can take to make the inequality true.
  3. The range of values can be represented on a number line.
  4. When a given endpoint of the range of values is included in the solution set, a closed circle is used, and when the endpoint is not included in the solution set, an open circle is used.

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