Operations on Inequalities

When you're performing algebraic operations on inequalities, it is important to conduct precisely the same operation on both sides in order to preserve the truth of the statement.

Each arithmetic operation follows specific rules:

Addition and Subtraction

Any value $c$ may be added to or subtracted from both sides of an inequality. That is to say, for any real numbers $a$, $b$, and $c$:

• If $a \leq b$, then $a + c \leq b + c$ and $a - c \leq b - c$.
• If $a \geq b$, then $a + c \geq b + c$ and $a - c \geq b - c$.

As long as the same value is added or subtracted from both sides, the resulting inequality remains true.

For example, consider the following inequality:

$12 < 15$

Let's apply the rules outlined above by subtracting 3 from both sides:

$\begin{aligned} 12 - 3 &< 15 - 3 \\ 9 &< 12 \end{aligned}$

This statement is still true.

Multiplication and Division

The properties that deal with multiplication and division state that, for any real numbers, $a$, $b$, and non-zero $c$:

If $c$ is positive, then multiplying or dividing by $c$ does not change the inequality:

• If $a \geq b$ and $c >0$, then $ac \geq bc$ and $\dfrac{a}{c} \geq \dfrac{b}{c}$.
• If $a \leq b$ and $c > 0 $, then $ac \leq bc$ and $\dfrac{a}{c} \leq \dfrac{b}{c}$.

If $c$ is negative, then multiplying or dividing by $c$ inverts the inequality:

• If $a \geq b$ and $c <0 $, then $ac \leq bc$ and $\dfrac{a}{c} \leq \dfrac{b}{c}$.
• If $a \leq b$ and $c < 0 $, then $ac \geq bc$ and $\dfrac{a}{c} \geq \dfrac{b}{c}$.

Take note that multiplying or dividing an inequality by a negative number changes the direction of the inequality. In other words, a greater-than symbol becomes a less-than symbol, and vice versa. 

To see these rules applied, consider the following inequality:

$5 > -3$

Multiplying both sides by 3 yields:

$\begin{aligned} 5 (3) &> -3 (3) \\ 15 &> -9 \end{aligned}$

We see that this is a true statement, because 15 is greater than 9.

Now, multiply the same inequality by -3 (remember to change the direction of the symbol because we're multiplying by a negative number):

$\begin{aligned} 5 (-3) &< -3 (-3) \\ -15 &< 9 \end{aligned}$

This statement also holds true. This demonstrates how crucial it is to change the direction of the greater-than or less-than symbol when multiplying or dividing by a negative number.

Solving Inequalities

Solving an inequality that includes a variable gives all of the possible values that the variable can take that make the inequality true. To solve an inequality means to transform it such that a variable is on one side of the symbol and a number or expression on the other side. Often, multiple operations are often required to transform an inequality in this way.

Addition and Subtraction

To see how the rules of addition and subtraction apply to solving inequalities, consider the following:

$x - 8 \leq 17$

First, isolate $x$:

$\begin{aligned} x - 8 + 8 &\leq 17 + 8 \\ x &\leq 25 \end{aligned}$

Therefore, $x \leq 25$ is the solution of $x - 8 \leq 17$. In other words, $x - 8 \leq 17$ is true for any value of $x$ that is less than or equal to 25.

Multiplication and Division

To see how the rules for multiplication and division apply, consider the following inequality:

$2x > 8$

Dividing both sides by 2 yields:

$\begin{aligned} \dfrac{2x}{2} &> \dfrac{8}{2} \\ x &> \dfrac{8}{2} \\ x &> 4 \end{aligned}$

The statement $x > 4$ is therefore the solution to $2x > 8$. In other words, $2x > 8$ is true for any value of $x$ greater than 4. 

Now, consider another inequality: 

$-\dfrac{y}{3} \leq 7$

Because of the negative sign involved, we must multiply by a negative number to solve for $y$. This means that we must also change the direction of the symbol:

$\begin{aligned} \displaystyle -3 \left( -\frac{y}{3} \right) &\geq -3 (7)\\ y &\geq -3 (7) \\ y &\geq -21 \end{aligned}$

Therefore, the solution to $-\frac{y}{3} \leq 7$ is $y \geq -21$. The given statement is therefore true for any value of $y$ greater than or equal to $-21$.

Example

Solve the following inequality: 

$3y - 17 \geq 19$

First, add 17 to both sides:

$\begin{aligned} 3y - 17 + 17 &\geq 19 + 17 \\ 3y &\geq 36 \end{aligned}$

Next, divide both sides by 3:

$\begin{aligned} \dfrac{3y}{3} &\geq \dfrac{36}{3} \\ y &\geq \dfrac{36}{3} \\ y &\geq 12 \end{aligned}$

Special Considerations

Note that it would become problematic if we tried to multiply or divide both sides of an inequality by an unknown variable. If any variable $x$ is unknown, we cannot identify whether it has a positive or negative value. Because the rules for multiplying or dividing positive and negative numbers differ, we cannot follow this same rule when multiplying or dividing inequalities by variables. Variables can, however, be added or subtracted from both sides of an inequality.