Mathematics » Equations and Inequalities » Solving Linear Inequalities

# Solving Linear Inequalities

## 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.