# 5.4 Linear inequalities

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## Investigation : inequalities on a number line

Represent the following on number lines:

1. $x=4$
2. $x<4$
3. $xâ‰¤4$
4. $xâ‰¥4$
5. $x>4$

A linear inequality is similar to a linear equation in that the largest exponent of a variable is 1. The following are examples of linear inequalities.

$\begin{array}{ccc}\hfill 2x+2& â‰¤& 1\hfill \\ \hfill \frac{2-x}{3x+1}& â‰¥& 2\hfill \\ \hfill \frac{4}{3}x-6& <& 7x+2\hfill \end{array}$

The methods used to solve linear inequalities are identical to those used to solve linear equations. The only difference occurs when there is amultiplication or a division that involves a minus sign. For example, we know that $8>6$ . If both sides of the inequality are divided by $-2$ , $-4$ is not greater than $-3$ . Therefore, the inequality must switch around, making $-4<-3$ .

When you divide or multiply both sides of an inequality by any number with a minus sign, the direction of the inequality changes. For this reason you cannot divide or multiply by a variable.

For example, if $x<1$ , then $-x>-1$ .

In order to compare an inequality to a normal equation, we shall solve an equation first. Solve $2x+2=1$ .

$\begin{array}{ccc}\hfill 2x+2& =& 1\hfill \\ \hfill 2x& =& 1-2\hfill \\ \hfill 2x& =& -1\hfill \\ \hfill x& =& -\frac{1}{2}\hfill \end{array}$

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

Now let us solve the inequality $2x+2â‰¤1$ .

$\begin{array}{ccc}\hfill 2x+2& â‰¤& 1\hfill \\ \hfill 2x& â‰¤& 1-2\hfill \\ \hfill 2x& â‰¤& -1\hfill \\ \hfill x& â‰¤& -\frac{1}{2}\hfill \end{array}$

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

As you can see, 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.

Solve for $r$ : $6-r>2$

1. $\begin{array}{c}\hfill -r>2-6\\ \hfill -r>-4\end{array}$
2. When you multiply by a minus sign, the direction of the inequality changes.

$r<4$

Solve for $q$ : $4q+3<2\left(q+3\right)$ and represent the solution on a number line.

1. $\begin{array}{ccc}\hfill 4q+3& <& 2\left(q+3\right)\hfill \\ \hfill 4q+3& <& 2q+6\hfill \end{array}$
2. $\begin{array}{ccc}\hfill 4q+3& <& 2q+6\hfill \\ \hfill 4q-2q& <& 6-3\hfill \\ \hfill 2q& <& 3\hfill \end{array}$
3. $\begin{array}{ccccc}\hfill 2q& <& 3\hfill & \mathrm{Divide both sides by}\phantom{\rule{2pt}{0ex}}2\hfill & \\ \hfill q& <& \frac{3}{2}\hfill & \end{array}$

Solve for $x$ : $5â‰¤x+3<8$ and represent solution on a number line.

1. $\begin{array}{ccc}\hfill 5-3â‰¤& x+3-3& <8-3\hfill \\ \hfill 2â‰¤& x& <5\hfill \end{array}$

## Linear inequalities

1. Solve for $x$ and represent the solution graphically:
1. $3x+4>5x+8$
2. $3\left(x-1\right)-2â‰¤6x+4$
3. $\frac{x-7}{3}>\frac{2x-3}{2}$
4. $-4\left(x-1\right)
5. $\frac{1}{2}x+\frac{1}{3}\left(x-1\right)â‰¥\frac{5}{6}x-\frac{1}{3}$
2. Solve the following inequalities. Illustrate your answer on a number line if $x$ is a real number.
1. $-2â‰¤x-1<3$
2. $-5<2x-3â‰¤7$
3. Solve for $x$ : $7\left(3x+2\right)-5\left(2x-3\right)>7$ Illustrate this answer on a number line.

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