Solving Inequalities Using the Subtraction and Addition Properties of Inequality

The Subtraction and Addition Properties of Equality state that if two quantities are equal, when we add or subtract the same amount from both quantities, the results will be equal.

Properties of Equality

\(\begin{array}{cccc}\mathbf{\text{Subtraction Property of Equality}}\hfill & & & \mathbf{\text{Addition Property of Equality}}\hfill \\ \text{For any numbers}\phantom{\rule{0.2em}{0ex}}a,b,\text{and}\phantom{\rule{0.2em}{0ex}}c,\hfill & & & \text{For any numbers}\phantom{\rule{0.2em}{0ex}}a,b,\text{and}\phantom{\rule{0.2em}{0ex}}c,\hfill \\ \begin{array}{cccc}\text{if}\hfill & \hfill a& =\hfill & b,\hfill \\ \text{then}\hfill & \hfill a-c& =\hfill & b-c.\hfill \end{array}\hfill & & & \begin{array}{cccc}\text{if}\hfill & \hfill a& =\hfill & b,\hfill \\ \text{then}\hfill & \hfill a+c& =\hfill & b+c.\hfill \end{array}\hfill \end{array}\)

Similar properties hold true for inequalities.

For example, we know that −4 is less than 2.
If we subtract 5 from both quantities, is the left side still less than the right side?
We get −9 on the left and −3 on the right.
And we know −9 is less than −3.
	The inequality sign stayed the same.

Similarly we could show that the inequality also stays the same for addition.

This leads us to the Subtraction and Addition Properties of Inequality.

Properties of Inequality

\(\begin{array}{cccc}\mathbf{\text{Subtraction Property of Inequality}}\hfill & & & \mathbf{\text{Addition Property of Inequality}}\hfill \\ \text{For any numbers}\phantom{\rule{0.2em}{0ex}}a,b,\text{and}\phantom{\rule{0.2em}{0ex}}c,\hfill & & & \text{For any numbers}\phantom{\rule{0.2em}{0ex}}a,b,\text{and}\phantom{\rule{0.2em}{0ex}}c,\hfill \\ \begin{array}{cccccc}& & \text{if}\hfill & \hfill a& <\hfill & b\hfill \\ & & \text{then}\hfill & \hfill a-c& <\hfill & b-c.\hfill \\ \\ & & \text{if}\hfill & \hfill a& >\hfill & b\hfill \\ & & \text{then}\hfill & \hfill a-c& >\hfill & b-c.\hfill \end{array}\hfill & & & \begin{array}{cccccc}& & \text{if}\hfill & \hfill a& <\hfill & b\hfill \\ & & \text{then}\hfill & \hfill a+c& <\hfill & b+c.\hfill \\ \\ & & \text{if}\hfill & \hfill a& >\hfill & b\hfill \\ & & \text{then}\hfill & \hfill a+c& >\hfill & b+c.\hfill \end{array}\hfill \end{array}\)

We use these properties to solve inequalities, taking the same steps we used to solve equations. Solving the inequality \(x+5>9\), the steps would look like this:

\(\begin{array}{cccccc}& & & \hfill x+5& >\hfill & 9\hfill \\ \text{Subtract 5 from both sides to isolate}\phantom{\rule{0.2em}{0ex}}x.\hfill & & & \hfill x+5-5& >\hfill & 9-5\hfill \\ \text{Simplify.}\hfill & & & \hfill x& >\hfill & 4\hfill \end{array}\)

Any number greater than 4 is a solution to this inequality.

Example

Solve the inequality \(n-\frac{1}{2}\le \frac{5}{8}\), graph the solution on the number line, and write the solution in interval notation.

Solution


Add \(\frac{1}{2}\) to both sides of the inequality.
Simplify.
Graph the solution on the number line.
Write the solution in interval notation.

Solving Inequalities Using the Subtraction and Addition Properties of Inequality