Linear inequalities

Linear formulas and equations: Linear equations and inequalities

Linear inequalities

Aside from knowing when two linear formulas are equal to each other, it can also be interesting to know when one linear formula is greater than the other. Therefore, we will take a look at how to solve linear inequalities.

Linear inequalities and graphs

We are interested in the case when the formula # \blue{y=3 \cdot x+1}# (solid) is less than the formula # \green{y=x-3}# (dashed). We write this as follows:

\[\blue{3 \cdot x+1} < \green{x-3} \]

This means we are interested in when the blue graph is below the green graph.

We see from the graph that that is the solution to the inequality: \[\orange{x\lt -2}\]

geogebra plaatje

Linear inequalities

A linear inequality describes when one linear formula is bigger/smaller than the other.

For writing down linear inequalities we use the following signs: #\lt#; #\leq#; #\gt# and #\geq#. Here #\lt# means less than; #\leq# less than or equal to #\gt# greater than, and #\geq# greater than or equal to.

The solution of a linear inequality is of the form: #x\lt a#; #x\leq a#; #x \gt a# or #x \geq a#.

Examples

\[\begin{array}{rcl}3 \cdot x+1&\lt&2 \\\\ 2 \cdot x -4 &\gt& -2 \cdot x +1 \\\\ 5 \cdot x&\leq& 3 \cdot x-1 \end{array}\]

Just as with linear equations, when working with linear inequalities we have a couple of steps we can perform, while keeping the inequality equivalent.

Equivalent inequalities

Two inequalities with exactly the same solution, are called equivalent. To indicate that two inequalities are equivalent, we can use the symbol #\Leftrightarrow#.

Example

\[\begin{array}{rcl}2x+2 \lt 1 &\Leftrightarrow&2x \lt -1 \end{array}\]

With the help of some rules, we can reduce inequalities to the form #x\lt a#; #a\leq a#; #x \gt a# or #x \geq a#. These rules are very similar to the reduction rules with equations. We will take a look at them one by one.

Two inequalities are equivalent if the same term is added to or subtracted from both sides.

Equivalent inequalities are usually written below each other, like shown on the right.

Example

\[\begin{array}{rcl}3x+4&\gt&5 \\ 3x+4-4&\gt& 5-4 \\ 3x&\gt&1 \end{array}\]

Two inequalities are equivalent if you multiply both sides by the same positive number, or divide by the same positive number.

Example

\[\begin{array}{rcl} 3x &\lt& 6 \\ \dfrac{3x}{3}&\lt& \dfrac{6}{3} \\ x&\lt& 2\end{array}\]

Two inequalities are equivalent if you multiply both sides by the same negative number and switch signs, or if you divide both sides by the same negative number and switch the sign.

By switching signs we mean that the sign becomes the opposite sign. Hence:

#\lt# becomes #\gt#	#\gt# becomes #\lt#
#\leq# becomes #\geq#	#\geq# becomes #\leq#

Example

\[\begin{array}{rcl}-3x &\lt& 6 \\ \dfrac{-3x}{-3}&\gt &\dfrac{6}{-3} \\ x&\gt& -2 \end{array}\]

With these rules we can reduce linear inequalities. In the examples below we will show how that is done.

Determine each #x# in #4\cdot x+2\geq 7-19\cdot x# .

Give your answer in one of the following forms with a suitable number for #a#.

#x \lt a#
#x\gt a#
#x\le a#
#x\ge a#

Simplify #a# as much as possible.

# x \geq {{5}\over{23}}#

#\begin{array}{rcl} 4x+2 &\geq & -19x+7\\&&\phantom{xxx}\blue{\text{given}} \\
23x &\geq& 5 \\&&\phantom{xxx}\blue{\text{subtracted }2-19x \text{ on both sides}}\\
x &\geq& {{5}\over{23}}\\&&\phantom{xxx}\blue{\text{divided by } 23 \text{ on both sides, because }23 \text{ is positive, the inequality sign does not change}}\\
\end{array}#

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