If you can solve linear equations, you can solve linear inequalities.
In this lesson, you will learn how similar the solving processes are. You will also learn the notations you need to use when graphing linear inequalities.
What Is a Linear Inequality?
A linear inequality is very similar to a linear equation. Recall that a linear equation has variables to the first degree only, and the variables are never squared, cubed, or taken to any other power.
Your linear equations might have looked something like these:
Linear inequalities look very similar. Instead of equal signs, though, linear inequalities have inequality symbols much like these:
To solve, we must isolate the variable by adding 5 to both sides. Then we divide both sides by 2 so that the variable is by itself.
The resulting solution is x ; 3, or all numbers less than 3.
Dividing by Negatives
There is one tricky aspect to solving linear inequalities, and that is when you are dividing or multiplying by a negative number. Whenever this is the case, the inequality symbol flips sides. Remember this – it’s an easily forgotten rule.For example:
The shaded region accounts for the inequality part.
Think about the linear inequality and what it says. It says that y is greater than x + 1. The region above the line gives you y values that are larger, and so that is what gets shaded.
Solving linear inequalities is very similar to solving linear equations.
The main difference is you flip the inequality sign when dividing or multiplying by a negative number. Graphing linear inequalities has a few more differences. When graphing on the number line, an open circle is for the symbols < and >. A solid circle is for the other two symbols. On the Cartesian graph, a dashed line is for the symbols < and >.
To represent the other two symbols, a solid line is used. The part that is shaded includes the values where the linear inequality is true.