Contents

# Inequalities Calculator + Online Solver With Free Steps

The **Inequality Calculator**Â is an online tool forÂ **evaluating inequalities**. It can be used to resolve a quadratic inequality and a linear inequality with oneÂ **unknown variable**.

Every time, the computations are done step-by-step, and precise results are provided.

## What Is an Inequalities Calculator?

TheÂ **Inequalities Calculator** is an online tool that determines the absolute value, rational, polynomial, quadratic, and linear inequalities.

**Inequalities are mathematical formulas that are used to make non-equal comparisons.**Â However, when both expressions are equal, the equality expression is employed.

Numerous mathematical problems compare the numbers using various inequalities, including less than ($<$), more significant than ($>$), less than or equal to ($\leq$), greater than or equal to ($\geq$), and not equal to ($\neq$).

**The less than and greater than inequalities**Â are the only ones of these that are regarded as rigorous inequalities.

## How To Use an Inequalities Calculator?

You can use theÂ **Inequalities Calculator** by following the detailed solution laid out below. The inequalities calculator will calculate the **value of the unknown variable**Â for the given expression.

### Step 1

Input the given data and enter the number of tails and directions in the specified fields on the Calculator’s layout.

### Step 2

PressÂ **the “Submit”**Â button to find theÂ **value of the unknownÂ **for the given expression, and also, the whole step-by-step solution for theÂ **Inequalities calculation**Â will be displayed.

## How Does an Inequalities Calculator Work?

The Inequalities Calculator works on the same principles as equation-based problem-solving, but because the comparison sign is present, it necessitates the following additional guidelines:

- The direction of inequality is altered by multiplying both sides by the same strictly negative real number:

if a$<$b and c$<$0, then a x c $>$ b x c

- The direction of inequality remains unchanged when both sides are multiplied by the same strictly positive real integer.

if a$<$b and c$>$0, then a x c $<$ b x cÂ

- When inequality is divided by the same strictly negative real number on both sides, the direction of the inequality is altered:

If a $<$ b and c $<$ 0, then a . c > b. c

- Dividing by the same strictly positive real number on each side of an inequality does not change the direction of the inequality:

If a $<$ b and c $>$ 0, then a . c < b . c

- A real number added to each side of an inequality, whether positive or negative, does not affect the inequality’s direction.

if a$<$b and c $\in R $ then a+c$<$b+c

- A real number that is the same on both sides of an inequality, whether positive or negative, does not affect the inequality’s direction.

if a$<$b and c $\in R $ then a-c$<$b-c

- An inequality’s direction is unaffected by squaring each of its positive sides:

if 0$<$a$<$b then $ a_2<b_2$

- An inequality’s direction changes when its negative sides are squared:

if a$<$b$<$0 then $ a_2>b_2$

- An inequality’s direction changes when each (non-zero) side is inverted:

if a$<$b then $ \frac{1}{a}> \frac{1}{b}$

It is also possible to merge several inequalities:

- Inequalities in the same direction are added from one member to the next:

if a$<$b and c$<$d, then a+c$<$b+d

- Inequalities in the same direction are multiplied member by member:

if 0$<$a$<$b and 0$<$c$<$d then a x c$<$b x d

### Operators in an Inequality

The Calculator accepts the following equation operators:

$ <= $ (less than or equal to)

$ > $ (strictly superior, greater than)

$ >= $ (greater than or equal)

$ <> $ or $ \neq $ (different, not equal)

The two inequality expressions, “x > 1” and “x^2 > x,” are not equivalent. This is because “x” in the inequality “x > 1” is bigger than 1.

However, if x is negative, then the inequality $ x^2 > x $ (which must be positive or zero) is always bigger than x. Thus we must account for this possibility.

In actuality, $ x > 1 $ or $ x < 0 $ is the whole answer to this inequality. Given that $ x^2 $ is always greater than x when x is negative, the second portion of the solution must be accurate.

### Principle of Solving an Inequality

- The Calculator applies the following ideas to solve inequality:
- It may increase or decrease both sides of an inequality by the same amount.
- Each component of inequality may be multiplied or divided by the same number.
- The direction of the inequality is reversed when this number is negative.
- When this number is positive, the perception of inequality is maintained.

## Solved Examples

Here are a few examples to better understand the working ofÂ **the Inequalities Calculator**.

### Example 1

**Solve 4x+3 $<$ 23?**

#### Solution

Given that

\[ 4x+3 < 23 \]

Subtract ‘-3’ from both sides.

\[ 4x+3 -3 < 23 â€“ 3 \]

\[ 4x < 20 \]

Divide ‘4’ into both sides

\[ \frac{4x}{4} < \frac{20}{4} \]

**x $<$ 5Â **

### Example 2

**Solve for c**

**\[ 3(x + c) – 4y \geq 2x – 5c \]**

#### Solution

Here, consider ‘c’ as variable and ‘x’ as constant.

\[ 3(x + c) – 4y \geq 2x – 5c \]

\[ 3x + 3c – 4y \geq 2x – 5c \]

\[ 3x – 2x – 4y \geq -5c -3c \]

\[ x – 4y \geq -8c \]

\[ 8c \leq 4y – x \]

\[ c \leq (4y – x)/ 8 \]

### Example 3

Solve the given inequality

\[ -2 < 6 – \frac{2x}{3} < 4 \]

#### Solution

First, let us multiply each part of the inequality by 3.

Since a positive number is being multiplied, the inequality does not change:

**-6 $<$ 6 – 2x $<$ 12**

Now after multiplying, subtract the number 6 on each side of the inequality:

**-12 $<$ -2x $<$ 6Â **

After that, divide each side by 2:

**-6 $<$ -x $<$ 3Â **

Lastly, multiply each side by âˆ’1. Since we are multiplying both the sides by aÂ **negative**Â number, the inequalities change the direction, which means that the less than symbol has changed into a greater than symbol as shown below:

**6 $>$ x $>$ -3Â **

And that is the solution

Though, just to be orderly, let’s switch the positions of numbers (and make sure the inequalities point correctly)

**Â -3 $<$ x $<$ 6Â **