Math Lesson 10.1.4 - Transformations made in Inequalities

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Welcome to our Math lesson on Transformations made in Inequalities, this is the fourth 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.

Transformations Made in Inequalities

The ultimate goal when dealing with inequalities is to isolate the variable, i.e. to write it alone on one of the sides (usually on the left), and after doing the necessary operations, obtain the simplest form of the inequality, that corresponds to the final solution.

We can complete the following transformation in inequalities for isolating the variable and therefore obtain an easier solution:

1. We can add or subtract the same number or expression from both sides of an inequality and still obtain an equivalent inequality to the original without any change in the inequality sign.
   For example, in the inequality
   x + 2 < 3
   we can subtract 2 from both sides to isolate x. In this way, we obtain
   x + 2 - 2 < 3 - 2
   x < 1
   This is the final (and simplest) version of the inequality x + 2 < 3 that was obtained by doing a single transformation.
2. We can multiply or divide both sides of an inequality by a positive number and still obtain an equivalent inequality to the original without any change in the inequality sign.
   For example, in the inequality
   3x ≤ 12
   we can divide both sides by 3 to isolate x. In this way, we obtain the simplified version of the original inequality:
   3x/3 ≤ 12/3
   x ≤ 4
4. We can multiply or divide both sides of an inequality by a negative number and still obtain an equivalent inequality to the original after changing the direction of the inequality sign.
   For example, in the inequality
   -4x > 20
   we can divide both sides by -4 to isolate x. However, we must also swap the direction of the inequality sign to obtain an equivalent inequality but in the simplified version. Thus,
   -4x/-4 > 20/-4
   x<-5
   Why is this so? Let's prove the third property of inequalities with the help of the other two. Thus, in the original inequality
   -4x > 20
   we can add both sides by 4x first, i.e.
   -4x + 4x>20 + 4x
   0 > 20 + 4x
   Now, we can remove 20 from both sides, i.e.
   0 - 20 > 20 + 4x - 20
   -20 ≥ 4x
   Moreover, we can divide both sides by 4 to isolate x:
   -20/4 ≥ 4x/4
   -5 ≥ x
   We can "read" the last mathematical sentence from right to left, as we did when dealing with double inequalities. In this way, we obtain
   x ≤ -5
   The last inequality is identical to the one obtained when using the third property of inequality. Hence, this property is confirmed as true.

It is evident from the last example that we can combine all the above properties in a single example until the desired result is obtained.

Example 3

Solve the following inequalities

1. 4x - 1 ≤ 19
2. 3 - 2x > 11
3. x/5 - 3 < 3x + 2

Solution 3

1. 4x - 1 ≤ 19
   4x - 1 + 1 ≤ 19 + 1
   4x ≤ 20
   4x/4 ≤ 20/4
   x ≤ 5
2. 3 - 2x > 11
   3 - 2x - 3 > 11 - 3
   -2x > 14
   -2x/-2 > 14/-2
   x < -7
3. x/5 - 3 < 3x + 2
   5 ∙ (x/5 -3 ) < 5 ∙ (3x + 2)
   5x/5 - 5 ∙ 3 < 5 ∙ 3x + 5 ∙ 2
   x - 15 < 15x + 10
   x - 15 - 15x < 15x + 10 - 15x
   -14x - 15 < 10
   -14x - 15 + 15 < 10 + 15
   -14x < 25
   -14x/-14 < 25/-14
   x > - 5/14

More Solving Linear Inequalities Lessons and Learning Resources

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1. Transformations Feedback. Helps other - Leave a rating for this transformations (see below)
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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