What are Equations and Inequations?

Equations and inequalities describe the relationship between two mathematical expressions.

What are Equations?

In math, an equation means that two algebraic expressions are equal. It is a mathematical sentence with an equal symbol in between the algebraic expressions.

Suppose that you have 5 chocolates in your cupboard. Your friend comes home and gives you 5 more chocolates. See if the total chocolates in your cupboard is the same as the total chocolates your friend gave you? Now the total number of chocolates is 5+5=10 and we call this an equation.

Algebraic Expression

An algebraic expression is made up of constants, variables, and operators. For example, $5x+1=11$ is an equation having two algebraic expressions connected by an equal sign (=). In algebra, the expressions are classified as variables, constants, and coefficients of variables. Here 1 and 11 are called constants because their value is fixed and definite, 5 is the value used together with x called the coefficient of x and x is the unknown variable. The variables and constants are combined using the operator’s addition (+) and multiplication$\left(\times \right)$. The left side of = is $5x+1$ and the right side is 11.

We read $5x+1=11$ as $5x+1$ is equal to 11.

What is an Inequality? 

Inequality means that two algebraic expressions are not connected by an equal sign. The two expressions are connected by the symbols $<,\le ,>\text{or}\ge$5<6 means 5 is strictly less than 6.

x$\le$y tells that the number x can be in low when compared to number y or the same as number y.

5>3 means 5 is strictly greater than 3.

x$\ge$y tells that the number x can be higher when comparing to number y or the same as number y.

By seeing some examples, we may get a clear understanding of equations and inequalities.

See if 4 is a solution to $2x+1=9$.

Is $2x+1=9$ an equation? Yes, it is an equation because the algebraic expressions $2x+1$and 9 are connected by an equal sign (=). In order to find if 4 is a solution we have to substitute 4 in the place of x. That is, put the value of x as 4. This gives, $\left(2\times 4\right)+1=9$.

Now we have to see if the left side of = in the equation is the same as the right side. If the value is the same on both sides, then 4 is a solution.

Left side is:

$\begin{array}{c}\left(2\times 4\right)+1=8+1\\ =9\end{array}$

Left side is 9. Right side is also 9. Therefore, we can conclude saying that 4 is a solution to the equation $2x+1=9$.

Is 5 a solution to this equation? Substitute the value of x as 5 and see if the left side and right side are the same. If it is not equal, then 5 is not a solution to $2x+1=9$. If you try, you will get $11\ne 9$.

What else can we say about the relationship between 11 and 9? 11 can be greater(>) / lesser(<) / higher or same as($\ge$)/ lower or same as($\le$)9.

Let us see if 3 a solution to $4x+2<x-2$.

Is $4x+2<x-2$ an equation? It is not an equation because the two algebraic expressions are connected by < sign. We have to check if $4x+2$ is lesser than $x-2$ when x=3.

Substituting x=3 in $4x+2<x-2$, we get

$\begin{array}{c}\left(4\times 3\right)+2<3-2\\ 12+2<3-2\\ 14<1\end{array}$

This is false because 14 is not less than 1. Therefore, 3 is not a solution $4x+2<x-2$.

Check if -2 is a solution to this inequality. (Answer: Yes, because -6<-4).

How to Solve an Equation? 

There are 4 different ways to solve a linear equation. They are summing up the values, taking the difference, multiplying and dividing. These operations should be carried on both sides of the equation. Let us start by solving simple equations.

1) Solve $x+4=0$.

We solve by finding x. We need x term alone on the left side of =. For this, take away 4 from both sides.

$\begin{array}{c}x+4=0\\ x+4-4=0-4\\ x+0=-4\\ x=-4\end{array}$

Therefore, -4 is the solution.

2) Solve $\frac{x}{3}=5$

To solve, we have to find x. We need x term alone on the left side of =. For this, multiply both sides by 3.  

$\begin{array}{c}\frac{x}{3}=5\\ \frac{x}{3}\times 3=5\times 3\\ x=15\end{array}$

3) Solve $8x=24$ .

We must have x alone on the left side of =, we get that by dividing 8 on both sides.

$\begin{array}{c}8x=24\\ \frac{8x}{8}=\frac{24}{8}\\ x=3\end{array}$

Let us try solving an equation involving all the above steps.

4) Solve $4x-14=\frac{x}{3}+8$

It takes 4 steps to solve the equation.

Step 1:  Add 14 on both sides.

$\begin{array}{l}4x-14+14=\frac{x}{3}+8+14\\ 4x=\frac{x}{3}+22\end{array}$

Step 2: Subtract $\frac{x}{3}$ on both sides.

$\begin{array}{c}4x-\frac{x}{3}=\frac{x}{3}+22-\frac{x}{3}\left(\text{the }\frac{x}{3}\text{terms on the right hand side will get cancelled}\right)\\ 4x-\frac{x}{3}=22\end{array}$

Step 3: Multiply by 3 on both sides.

$\begin{array}{c}3\times \left(4x-\frac{x}{3}\right)=3\times 22\\ 12x-\frac{3x}{3}=66\end{array}$

Step 4: Simplify to get the value of x.

$\begin{array}{c}12x-x=66\\ 11x=66\left(\text{divide by 11 on both sides}\right)\\ \frac{11x}{11}=\frac{66}{11}\\ x=6\end{array}$

Therefore, x=6 is the solution to the equation $4x-14=\frac{x}{3}+8$.

Try to solve the equation 4x+7=1+2x.

How to Solve an Inequality?

Now we shall see how to solve the inequalities. To solve we must bring the variable(x, y, or any variable) to the left side of the inequality (<,>,$\le ,\ge$) and the constants on the right side. There are some things that we have to keep in mind for solving.

Addition and Subtraction

We can add a positive number or a negative number on both sides of =. (Adding a number less than 0 means subtracting a positive number).

Multiplication and Division

1. We can multiply or divide the inequality by a positive number on both sides. (Multiplying or dividing a positive number will not affect the inequality).

2.When we multiply or divide the inequality by a negative number on both sides we must reverse the inequality by changing > as < (higher than becomes lower than), < as > (lower than becomes higher than), $\le$ as $\ge$ (lower than or same as to higher than or same as) and $\ge \text{as }\le$ (higher than or same as to lower than or same as).

All the above operations should be carried on both sides of the inequality.

Let us see some problems.

1) $x-3<5$.

In this case we have to add 3 on both sides to get x on the left side.

$\begin{array}{c}x-3+3<5+3\\ x<8\end{array}$

2) $2x+5<x-3$.

We can solve it in two steps.

Step 1: We have 2x+5 on the left side so we must subtract x from 2x on the left side. But this has to be done on both sides simultaneously. Therefore, subtract x on both sides.

$\begin{array}{c}2x+5-x<x-3-x\\ x+5<-3\end{array}$

Step 2: Subtract 5 on both sides.

$\begin{array}{c}x+5-5<-3-5\\ x<-8\end{array}$

3) $\frac{x+4}{7}\ge 1$ .

Two steps are involved to solve.

Step 1: Multiply by 7 on both sides of the inequality.

$\begin{array}{c}\left(\frac{x+4}{7}\right)\times 7\ge 1\times 7\\ \left(x+4\right)\ge 7\left(\text{on the left hand side }\frac{7}{7}=1\right)\end{array}$

Step 2: Subtract 4 on both sides.

$\begin{array}{c}x+4-4\ge 7-4\\ x\ge 3\end{array}$

4) $-4x\le 16$

Divide -4 on both sides to get on the left of the inequality.

$\begin{array}{c}\frac{-4x}{-4}\ge \frac{16}{-4}\\ x\ge -4\end{array}$

Formula

The most commonly used formula to form equations is the distance formula. Here, ‘d’ represents the distance, ‘r’ represents the rate and ‘t’ denotes the time.$d=r\cdot t$

Common Mistakes

It is possible that we may add or multiply on the left side alone or right side alone. But this should be done on both sides.

Use the same operation on both sides (if you add on the left side, you must add and not subtract on the right side of the equation or inequality).

Context and Applications:

It is used in K-12 curriculum, undergraduate and postgraduate mathematics, and asked in entrance examinations.