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Inequalities Calculator + Online Solver With Free Steps
The Inequality Calculator is an online tool for evaluating inequalities. It can be used to resolve a quadratic inequality and a linear inequality with one unknown variable.
Every time, the computations are done step-by-step, and precise results are provided.
What Is an Inequalities Calculator?
The 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. 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 are the only ones of these that are regarded as rigorous inequalities.
How To Use an Inequalities Calculator?
You can use the Inequalities Calculator by following the detailed solution laid out below. The inequalities calculator will calculate the value of the unknown variable 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
Press the “Submit” button to find the value of the unknown for the given expression, and also, the whole step-by-step solution for the Inequalities calculation 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 c
- 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 of the Inequalities Calculator.
Example 1
Solve 4x+3 $<$ 23?
Solution
Given that
\[ 4x+3 < 23 \]
Subtract ‘-3’ from both sides.
\[ 4x+3 -3 < 23 – 3 \]
\[ 4x < 20 \]
Divide ‘4’ into both sides
\[ \frac{4x}{4} < \frac{20}{4} \]
x $<$ 5
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 $<$ 6
After that, divide each side by 2:
-6 $<$ -x $<$ 3
Lastly, multiply each side by −1. Since we are multiplying both the sides by a negative 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 $>$ -3
And that is the solution
Though, just to be orderly, let’s switch the positions of numbers (and make sure the inequalities point correctly)
-3 $<$ x $<$ 6