Solving Linear Inequalities

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10.1Solving Linear Inequalities

In this Math tutorial, you will learn:

  • 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?

Introduction

How do you express the fact that two quantities A and B are not equal? If A has more items than B, how do you express this fact in symbols?

Imagine you have to score at least 80 points in an exam. How do you express this fact in math symbols?

So far we have discussed linear and quadratic equations solved through definite methods as well as other types of equations solved through iterative methods. In all cases, the objective was to find the value(s) of variable(s) that make the equation true, i.e. that make the left side equal to the right side.

In this chapter, we will discuss situations where the left side is not equal to the right one. Such mathematical sentences are known as inequalities.

This guide in particular focuses on linear inequalities, as there are the simplest type of inequalities. We will explain how to solve linear equalities using by different approaches. All situations discussed in this guide are illustrated with examples, as always.

What are Inequalities?

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".

For example, we read the inequality 3 - 2x ≥ 5y as "3 - 2x is greater than or equal to 5y."

If an inequality bears one of the two double signs written above, it is sufficient that only one of the conditions meets to make the inequality true. For example, the inequality 5 ≤ 7 is true because one of the conditions, 5 < 7 is met.

Solving an Inequality (with one variable)

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 a finite number of values as solutions, as usually happens in equations, but a set of infinite values instead.

For example, if after doing all necessary operations we obtain x > 3 as the final (simplest) version of inequality, this means all values of the variable x that are greater than 3 are solutions of the given inequality. We say all these values belong to the solution set of that inequality.

Example 1

Which of the following numbers do not belong to the solution set of the inequality

x ≥ 7
  1. 8
  2. 7
  3. 6
  4. 5

Solution 1

  1. The number 8 belongs to the solution set of the inequality x ≥ 7 as if we write 8 ≥ 7, this inequality becomes true.
  2. The number 7 belongs to the solution set of the inequality x ≥ 7 as if we write 7 ≥ 7, this inequality becomes true.
  3. The number 6 does not belong to the solution set of the inequality x ≥ 7 as if we write 6 ≥ 7, this inequality becomes false.
  4. The number 5 does not belong to the solution set of the inequality x ≥ 7 as if we write 5 ≥ 7, this inequality becomes false.

Double Inequalities and the Symbols used to Express Solution Sets

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. For example, 1 < x < 7 is a double inequality, as it expresses the inequality "the values of the variable x must be between 1 and 7 without including these limit values", which can be written in two separate parts, such as "x > 1" and "x < 7", where the solution set must include only numbers that make both these individual inequalities true.

Example 2

Write the following individual inequalities as a double inequality

  1. x ≥ -2 and x < 9
  2. x > 0 and x < 5

Solution 2

We must focus on the variable and read the double inequalities from there. The variable, however, must be in the middle of the sentence that contains the double inequality. In this way, we must invert laterally the direction of the first inequality. Hence, we obtain

  1. x ≥ -2 and x < 9 → -2 ≤ x < 9
  2. x > 0 and
    x < 5 → 0 < x < 5

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/312/3
    x ≤ 4
  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.
    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:
    -20/44x/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/420/4
    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

Intervals and Segments

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

Math Tutorials: Solving Linear Inequalities Example

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/-412/-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."
  2. 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."
  3. 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."
  4. 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
    -6/33x/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."

Solving Linear Inequalities in Two Variables

When dealing with linear equations with two variables, we explained that a single linear equation gives an infinity of number pairs that are all solutions for the given equation. We cannot expect anything else when dealing with linear inequalities as they usually contain more possible solutions than the corresponding equations. In other words, 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, which we will explain in tutorial 10.3. 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. For example, if we have to find the solution set of the linear inequality

3x - y < 1

if the variable x takes the values from the segment [2, 7], we must solve two linear inequalities with one variable like those discussed in the previous paragraphs, where the variable x is replaced by the two limit values of the given segment. In this way, the two linear inequalities are solved only for y.

Thus, for x = 2, we obtain

3x - y < 1
3 ∙ 2 - y < 1
6 - y < 1
6 - y - 6 < 1 - 6
-y < -5
-y ∙ (-1) < -5 ∙ (-1)
y > 5

and for x = 7, we obtain

3x - y < 1
3 ∙ 7 - y < 1
21 - y < 1
21 - y - 21 < 1 - 21
-y < -20
-y ∙ (-1) < -20 ∙ (-1)
y > 20

Thus, since the values of y must be limited between the values found for the two limit values of x, we obtain for the solution set of the inequality the values of y that extend from 5 and up for the minimum value of x, to 20 and up for the maximum value of x, without including these limit values.

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. Let's take an example to clarify this point.

Example 5

Check whether the number pairs (3, 1), (-3, 4) and (0, 2) belong to the solution set of the inequality

1 - 4x ≥ 2y + 3

Solution 5

First, let's write the inequality in such a way as to isolate y on the left side and express all the rest of the terms on the right side of the inequality symbol. Thus,

1 - 4x ≥ 2y + 3

Looking at this inequality from right to left yields

2y + 3 ≤ 1 - 4x

Applying the properties of inequalities yields

2y + 3 - 3 ≤ 1 - 4x - 3
2y ≤ -4x - 2
2y/2-4x/2 - 2/2
y ≤ -2x - 1

Now, let's check whether the given number pairs are solutions for the above inequality or not. Thus, for the number pair (3, 1) (x = 3 and y = 1) we have

1 ≤ -2 ∙ 3 - 1
1 ≤ -6 - 1
1 ≤ -7 (false)

Hence, the number pair (3, 1) is not a solution for the given linear inequality.

As for the number pair (-3, 4) (x = -3 and y = 4), we have

4 ≤ -2 ∙ (-3) - 1
4 ≤ 6 - 1
4 ≤ 5 (true)

Hence, the number pair (-3, 4) is a solution for the given linear inequality.

Last, for the number pair (0, 2) (x = 0 and y = 2), we obtain

2 ≤ -2 ∙ 0 - 1
2 ≤ 0 - 1
2 ≤ -1 (false)

Hence, the number pair (0, -4) is not a solution for the given linear inequality.

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