Solving Linear Inequalities - Revision Notes

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10.1Solving Linear Inequalities


In these revision notes for Solving Linear Inequalities, we cover the following key points:

  • What are inequalities? How do they differ from equations?
  • What are the symbols used to express the relationships between quantities in inequalities?
  • What does 'solving an inequality' mean? When is an inequality considered solved?
  • Can we obtain a definite number of values when solving an inequality? Why?
  • What are double inequalities? How do we express them as single inequalities?
  • How do we use math symbols to express the solution set of an inequality?
  • What kind of transformations are allowed in inequalities?
  • What are intervals and segments? Why are they important in inequalities?
  • How do we solve linear inequalities in two variables?
  • How do we check whether a number pair is a solution for a linear inequality in two variables or not?

Solving Linear Inequalities Revision Notes

Unlike in equations, where the left part is always equal to the right one, inequalities are mathematical sentences composed of two mathematical expressions, where the expression on the left side is not equal to that on the right side.

We use four symbols to represent inequalities. They are:

" > ", which means "greater than";

" < ", which means "smaller than";

" ", which means "greater than or equal to"; and

" ", which means "smaller than or equal to".

If an inequality bears a double sign, it is sufficient that only one of the conditions meets to make it true.

Solving an inequality means finding the set of values that make the inequality true. This means that when solving an inequality, we don't have to find anymore a finite number of values as solutions, as usually happens in equations, but a set of infinite values instead.

Sometimes we may encounter some special inequalities that contain two inequality symbols, consisting of an algebraic expression in-between, and numbers aside from these inequality symbols. In such cases, we are dealing with double inequalities. We can also write them as two separate single inequalities, where the solution set must include only numbers that make both these individual inequalities true.

We can do 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.
  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.
  3. 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.

We can combine all the above properties in a single inequality until the desired result is obtained.

There are some special symbols that represent sets of numbers determined by inequalities. They are written in the table below.

Math Tutorials: Solving Linear Inequalities Example

A linear inequality with two variables is solved by finding all number pairs that make it true. The number of possible solutions of the linear inequality

y > mx + n

is much greater than that of the corresponding equation

y = mx + n

despite both having an infinite number of possible solutions.

Linear inequalities with two variables are better understood when using the graph method. However, we can use analytical methods to solve linear inequalities as well. Just one thing to remember: we have to choose a range of allowed values for the dependent variable x and based on this range calculate the corresponding range of values for the dependent variable y.

We can check whether a certain number pair belongs to the solution set of a linear inequality or not by substituting the values in the inequality. In this way, after doing all operations, we see whether the final version of the simplified inequality is true or not.

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  5. Continuing learning inequalities - read our next math tutorial: Quadratic Inequalities

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