Math Lesson 10.1.5 - Intervals and Segments

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Welcome to our Math lesson on Intervals and Segments, this is the fifth lesson of our suite of math lessons covering the topic of Solving Linear Inequalities, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Intervals and Segments

There are some special symbols that represent sets of numbers determined by inequalities. Let's explain them through the following table.

Remarks!

1. We use the interval symbols "(" for minus infinity and ")" for plus infinity as it is impossible to find an exact number that represents infinity.
2. When the two limit values a and b are included in an inequality, we are dealing with one of the first four cases shown in the table, while when there is only one limit value present in an inequality, we are dealing with the rest of cases (without the last one).

Example 4

Solve the following inequalities by giving the final answer in set symbols.

1. 1 - 4x ≥ 13
2. 5x - 6 < 9
3. 3 + 2x < 4x - 7 < 2x + 11
4. -2 ≤ 3x + 4 < 16

Solution 4

1. Using the properties of inequalities yields
   1 - 4x ≥ 13
   1 - 4x - 1 ≥ 13 - 1
   -4x ≥ 12
   -4x/-4 ≥ 12/-4
   x ≤ -3
   When expressed in set symbols, this solution becomes:
   x ϵ (-∞, -3]
   where the symbol "ϵ" means "is an element of the set" or "belongs to the set".
   Hence, we read the result as: "The solution set that contains all values extending from minus infinity to -3, including this limit value."
Again, using the properties of inequalities yields
5x - 6 < 9
5x - 6 + 6 < 9 + 6
5x < 15
5x/5 < 15/5
x < 3
When expressed in set symbols, this solution becomes:
x ϵ (-∞, 3)
We read this result as: "The solution set contains all values extending from minus infinity to 3, without including this limit value."
First, we have to make operations on all sides of this double inequality to isolate the variable x. Thus, we have
3 + 2x < 4x-7 < 2x + 11
3 + 2x - 2x < 4x - 7 - 2x < 2x + 11 - 2x
3 < 2x - 7 < 11
3 + 7 < 2x - 7 + 7 < 11 + 7
10 < 2x < 18
10/2/ < 2x/2 < 18/2
5 < x < 9
When expressed in set symbols, this solution becomes:
x ϵ (5, 9)
We read this result as: "The solution set contains all values extending from 5 to 9, without including these two limit values."
Again, we have a double inequality, so we have to comolete operations in all three parts of it. Thus, applying the properties of inequalities yields
-2 ≤ 3x + 4 < 14
-2-4 ≤ 3x + 4 - 4 < 16 - 4
-6 ≤ 3x < 12
-6/3 ≤ 3x/3 < 12/3
-2 ≤ x < 4
When expressed in set symbols, this solution becomes:
x ϵ [-2, 4)
We read this result as: "The solution set contains all values extending from -2 to 4, including -2 but not 4."

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2. Inequalities Math tutorial: Solving Linear Inequalities. Read the Solving Linear Inequalities math tutorial and build your math knowledge of Inequalities
3. Inequalities Video tutorial: Solving Linear Inequalities. Watch or listen to the Solving Linear Inequalities video tutorial, a useful way to help you revise when travelling to and from school/college
4. Inequalities Revision Notes: Solving Linear Inequalities. Print the notes so you can revise the key points covered in the math tutorial for Solving Linear Inequalities
5. Inequalities Practice Questions: Solving Linear Inequalities. Test and improve your knowledge of Solving Linear Inequalities with example questins and answers
6. Check your calculations for Inequalities questions with our excellent Inequalities calculators which contain full equations and calculations clearly displayed line by line. See the Inequalities Calculators by iCalculator™ below.
7. Continuing learning inequalities - read our next math tutorial: Quadratic Inequalities

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