# Equations and Inequalities

**Equations**

**Equations**

**Equations** are a mathematical statement that shows **equality** between two **expressions**. The equality is denoted by an equals "=" sign. We often use equations to solve for an unknown variable.

Here are a few examples of equations:

x = 14

3p + 1 = 4

2x = 4y

ax = b/2

We solve for unknown variables in equations by isolating the variable on one side of the equation with all the known terms or constants on the other side. This can be done by adding, subtracting, multiplying, and/or dividing both sides of the equation by a fixed value.

**Example 1.** Solve for *x* in *3x + 1 = 7.*

**Example 1.**Solve for

*x*in

*3x + 1 = 7.*

Our first step to approaching this problem is to isolate *x*. We can do this by first isolating *3x* by subtracting both sides of the equation by 1. This way the 1 and -1 will cancel each other out, leaving us with only *3x* on the left side.

We have *3x + 1 = 7*

* ** ** ** ** ** ** ** ** ** *-->* 3x + 1 - 1 = 7 - 1*

* ** ** ** ** ** ** ** ** ** *-->* 3x + 0 = 6*

* ** ** ** ** ** ** ** ** ** *-->* 3x = 6*

Next we divide both sides of the equation by 3 to isolate *x*. After dividing, we have solved for *x*.

We have *3x = 6*

* ** ** ** ** ** ** ** ** ** *-->* 3x/3 = 6/3*

* ** ** ** ** ** ** ** ** ** *-->* 1x = 2*

* ** ** ** ** ** ** ** ** ** *-->* ***x = 2**

**Example 2.** Solve for *x* in *5x - 6 = 4x -7.*

**Example 2.**Solve for

*x*in

*5x - 6 = 4x -7.*

We approach this problem the same way as the previous example, except we have a few more terms to move around. This time, we will add 6 to both sides of the equation to the -6 and 6 on the left side will cancel out.

We have *5x - 6 = 4x - 7*

* ** ** ** ** ** ** ** ** ** *-->* 5x - 6 + 6 = 4x - 7 + 6*

* ** ** ** ** ** ** ** ** ** *-->* 5x + 0 = 4x - 1*

* ** ** ** ** ** ** ** ** ** *-->* 5x = 4x - 1*

Next, we subtract *4x* from both sides of the equation to isolate *x* on the left side. We will have solved for *x*.

We have *5x = 4x - 1*

* ** ** ** ** ** ** ** ** ** *-->* 5x - 4x = 4x - 4x - 1*

* ** ** ** ** ** ** ** ** ** *-->* x = 0 - 1*

* ** ** ** ** ** ** ** ** ** *-->* ***x = -1**

**Inequalities**

**Inequalities**

An algebraic inequality is a **non-equal** relation between two expressions. We manipulate terms similar to how we solve algebraic equations, but the answer gives us a **range** of values that satisfy the inequality.

Examples of inequalities include:

x >= 14

3p + 1 < 4

2x <= 4y

ax > b/2

However, when multiplying or dividing an inequality by a negative value on both sides, we must flip the sign of the inequality. "<" becomes ">", "<=" becomes ">=", and vice versa.

Here is what happens when we multiply both sides of an inequality by a negative number.

Here is an example of how flipping signs work. This applies for multiplication as well.

**Example 3.** Solve for *x* in the inequality *2x - 14 >= 6.*

**Example 3.**Solve for

*x*in the inequality

*2x - 14 >= 6.*

We can solve this inequality the same way we solve equations. We first isolate *2x* by adding 14 to both sides of the inequality. -14 and 14 will cancel out, leaving us with an inequality between *2x* and a constant.

We have *2x - 14 >= 6*

* ** ** ** ** ** ** ** ** ** *-->* 2x - 14 + 14 >= 6 + 14*

* ** ** ** ** ** ** ** ** ** *-->* 2x + 0 >= 20*

* ** ** ** ** ** ** ** ** ** *-->* 2x >= 20*

We then divide both sides with two, leaving *x* with a coefficient of 1, which means we have solved the inequality.

We have *2x >= 20*

* ** ** ** ** ** ** ** ** ** *-->* 2x/2 >= 20/2*

* ** ** ** ** ** ** ** ** ** *-->* 1x >= 10*

* ** ** ** ** ** ** ** ** ** *--> **x >= 10**

We can see that the final inequality represents a range of values of *x* which makes the original equation true. Out unknown, *x,* can be any real number greater than or equal to 10.

**Example 4.** Solve for *x* in the inequality *-13x + 4 < 7x - 9.*

**Example 4.**Solve for

*x*in the inequality

*-13x + 4 < 7x - 9.*

Let's first start by subtracting 4 from both sides of the inequality.

We have *-13x + 4 < 7x - 9*

* ** ** ** ** ** ** ** ** ** *-->* -13x + 4 - 4 < 7x - 9 - 4*

* ** ** ** ** ** ** ** ** ** *-->* -13x + 0 < 7x - 13*

* ** ** ** ** ** ** ** ** ** *-->* -13x < 7x - 13*

We can then isolate *x* with a coefficient by subtracting *7x* from both sides of the inequality.

We have *-13x < 7x - 13*

* ** ** ** ** ** ** ** ** ** *-->* -13x - 7x < 7x - 13 - 7x*

* ** ** ** ** ** ** ** ** ** *-->* -20x < 0 - 13*

* ** ** ** ** ** ** ** ** ** *-->* -20x < -13*

Now we can divide both sides of the inequality by -20 to isolate *x*. When dividing by a negative number, we must flip the sign from less than "<" to greater than ">".

We have *-20x < -13*

* ** ** ** ** ** ** ** ** ** *-->* -20x/-20 < -13/-20*

* ** ** ** ** ** ** ** ** ** *-->* 1x ***>*** 13/20*

* ** ** ** ** ** ** ** ** ** *-->* ***x > 13/20**