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Inequalities are mathematical statements that establish a relation between two expressions by comparing them. We will discuss the operations that we can use to solve inequalities. We will also learn the significance of symbols like <, >, ≥, ≤, and ≠ in inequalities....Read MoreRead Less

- What is the Method applied to Solve Multi-Step Inequalities?
- What are the Properties of Inequality
- Addition property of inequalities
- Subtraction Property of Inequalities
- Multiplication and Division Property 1
- Multiplication and Division Property 2
- How do we Solve Multi-Step Inequalities? What are Two-Step Inequalities?
- Solved Examples
- Frequently Asked Questions

Inequalities characterize the relationship between two values that are not equal in mathematics. An inequality denotes a lack of equality. When two values are not equal, we use the “not equal sign \(‘(\neq)’\) in most cases. Different inequalities are used to compare the values, whether they are less than or larger than. Inequalities are similar to equations. Equations have an equals sign, inequalities have an inequality symbol. It denotes that two sides or values may or may not be equal.

Inequalities can be expressed using algebra tiles or graphs. But in either case, the inequality needs to be simplified first, and to simplify the inequality, there are a few properties that need to be considered.

Given below are the different properties of inequalities. Note that every property here is applicable for inequalities involving the symbols, “<, >, \(\leq~\text{and}~\geq\)”.

If both the sides of an inequality are added with the same number, the inequality remains the same.

**For example:**

Numbers:

– 7 <8

– 7 + 5 < 8 + 5

– 2 < 13

**Algebra:**

If a < b, then a + c < b + c

If a > b, then a + c > b + c

If both the sides of an inequality are subtracted with the same number, the inequality remains the same.

**Numbers:**

13 > 9

13 – 7 > 9 – 7

6 > 2

**Algebra:**

If a < b, then a – c < b – c

If a > b, then a – c > b – c

When an inequality is multiplied or divided by the same positive number on either side of the inequality, the inequality remains the same.

**Numbers:**

12 > 8

12 x 2 > 8 x 2

24 > 16

8 < 12

\(\frac{8}{4}\) < \(\frac{12}{4}\)

2 < 3

**Algebra:**

If a < b, and c is positive, then a x c < b x c and \(\frac{a}{c}~ < ~\frac{b}{c}\)

\(\frac{a}{c}\) < \(\frac{b}{c}\)

If a > b and c is positive, then a x c > b x c and \(\frac{a}{c}~ > ~\frac{b}{c}\)

If either side of an inequality is multiplied or divided by the same negative number the direction of the inequality symbol should be reversed for the inequality to remain true.

– 5 < 10

– 2 x -5 > – 2 x 10

10 > – 20

5 > – 10

\(\frac{5}{(-5)}~~<~~-~\frac{10}{(-5)}\)

– 1 < 2

If a < b and c is negative, then a x c > b x c and \(\frac{a}{c}~ > ~\frac{b}{c}\)

If a > b and c is negative, then a x c < b x c and \(\frac{a}{c}~ < ~\frac{b}{c}\)

In order to represent an inequality graphically, it must first be simplified such that only the variable and a term remain. As the name suggests, two-step inequalities can be simplified to its simplest form in two steps.

To do this, undo the addition or subtraction first, then the multiplication or division, utilizing inverse operations to solve a two-step inequality.

Let’s take a look at an example using algebra tiles.

2x + 2 \(\geq\) – 2

The first step is to push the variable to one side and the numbers to the other. So, we subtract two from both sides.

2x + 2 – 2 \(\geq\) – 2 – 2

2x \(\geq\) – 4

To simplify the inequality further both sides should be divided by 2.

\(\frac{2x}{2}~\geq~-~\frac{4}{2}\)

x \(\geq\) -2

When representing this inequality on a graph, we get the following.

When representing these inequalities on a number line there are a few things to remember. Since the representation is not a single term, the solution will not be one point on the number line. It can stretch across a large number of values. The starting point of the inequality is represented by a dot, which is either shaded or not shaded and the remaining part of the number line is also shaded according to the situation.

**Simplify the following inequalities and plot the solutions on a number line.**

**Example 1:**

6x + 10 \(\geq\) 22

**Solution:**

6x + 10 \(\geq\) 22

6x + 10 – 10 \(\geq\) 22 – 10

6x \(\geq\) 12

6x x \(\frac{1}{6}~\geq\) 12 x \(\frac{1}{6}\)

x \(\geq\) 2

**Example 2:**

2 > 7 – \(\frac{5}{4}\)h

**Solution:**

2 > 7 – \(\frac{5}{4}\)h

2 – 7 > 7 – 7 – \(\frac{5}{4}\)h

– 5 > – \(\frac{5}{4}\)h

– 5 > – \(\frac{5}{4}\)h

5 < \(\frac{5}{4}\)h

5 x \(\frac{4}{5}\) < \(\frac{5}{4}\) x \(\frac{4}{5}\)h

4 < h

h > 4

**Simplify the following inequalities using algebraic tiles. Also, write the numeric form of the question and the final solution.**

**Example 3:**

**Solution:**

This represents: x – 4 \(\geq\) 8

Hence the solution is: x \(\geq\) 12

**Example 4:**

**Solution:**

This represents: 3x – 2 \(\geq\) 10

**Hence the solution is: **x \(\geq\) 4

**Example 5**

A race car was taken for a test drive and the driver figured that there was something wrong with the car if he went beyond 100 miles per hour. To be safe, he drives at a certain speed which is 25 miles per hour below the danger limit. He can’t stop in the middle of the road as he is far away from his starting point and his phone is out of charge. Represent this situation which shows at what point does the speed get dangerous from the current speed. Draw a graph representing the same as well.

**Answer:**

Let’s assume that the car moves at x miles per hour. The inequality depicting the situation looks like this.

x + 25 < 100

x + 25 – 25 < 100 – 25

x < 75

Frequently Asked Questions on Solving Inequalities

Inequalities that require two steps to solve are known as two-step inequalities. This means that to solve the inequality, you must add, subtract, multiply, or divide twice. To answer each two step inequality, you must first add or subtract, then multiply or divide the inequality.

The properties of inequalities are certain rules and restrictions that should be kept in mind when an inequality has to be simplified further to obtain the solution of a problem.