## Solving Linear Inequalities

A linear inequality is similar to a linear equation in that the largest exponent of a variable is \(\text{1}\). The following are examples of linear inequalities.

\begin{align*} 2x + 2 & \le 1 \\ \cfrac{2 – x}{3x + 1} & \ge 2 \\ \cfrac{4}{3}x – 6 & < 7x + 2 \end{align*}

The methods used to solve linear inequalities are similar to those used to solve linear equations. The only difference occurs when there is a multiplication or a division that involves a minus sign. For example, we know that \(8>6\). If both sides of the inequality are divided by \(-\text{2}\), then we get \(-4>-3\), which is not true. Therefore, the inequality sign must be switched around, giving \(-4<-3\).

In order to compare an inequality to a normal equation, we shall solve an equation first.

Solve \(2x + 2 = 1\):

\begin{align*} 2x + 2 & = 1 \\ 2x & = 1 – 2 \\ 2x & = -1 \\ x & = -\cfrac{1}{2} \end{align*}

If we represent this answer on a number line, we get:

Now let us solve for \(x\) in the inequality \(2x + 2 \le 1\):

\begin{align*} 2x + 2 & \le 1 \\ 2x & \le 1 – 2 \\ 2x & \le -1 \\ x & \le -\cfrac{1}{2} \end{align*}

If we represent this answer on a number line, we get:

We see that for the equation there is only a single value of \(x\) for which the equation is true. However, for the inequality, there is a range of values for which the inequality is true. This is the main difference between an equation and an inequality.

**Remember:** when we divide or multiply both sides of an inequality by a negative number, the direction of the inequality changes. For example, if \(x<1\), then \(-x>-1\). Also note that we cannot divide or multiply by a variable.

### Fact:

The following video provides an introduction to linear inequalities.