How to Solve Inequalities Graphically

Here you’ll learn how to solve inequalities graphically. When you have an inequality, you can think of it as two functions linked together by an inequality sign. That is, you can draw the graph of the left-hand side of the inequality, and draw the graph of the right-hand side of the inequality, by putting “$y=$” in front of the expressions.

Example 1

Solve the inequality $2x+4>0$ graphically

1.
Draw the line of the left-hand side of the inequality, $y=2x+4$, in a coordinate system.
2.
Draw the line of the right-hand side of the inequality, $y=0$, in a coordinate system.
3.
Mark the point where the graphs intersect. The graphs look like this:

4.
Read off the $x$-value at the point of intersection. From the figure, you see that the graphs meet at $x=-2$. You also see that the function $y=2x+4$ is over $y=0$ to the right of $-2$. The answer is therefore $x>-2$.

Rule

GraphicalSolutionsofInequalities

1.
Draw the graph of the left-hand side of the inequality in a coordinate system.
2.
Draw the graph of the right-hand side of the inequality in the same coordinate system.
3.
Mark where the graphs intersect.
4.
Read the $x$-value of the points where the graphs intersect.

Example 2

Solve the inequality ${x}^{2}\ge 4$ graphically

1.
Draw the graph of the left-hand side of the inequality in a coordinate system, $y={x}^{2}$.
2.
Draw the graph of the right-hand side of the inequality in the same coordinate system, $y=4$.
3.
Mark where the graphs intersect. The graphs look like this:

4.
Read off the $x$-values at the two points of intersection. From the figure you can see that the graphs intersect when ${x}_{1}=-2$ and ${x}_{2}=2$.

The parabola $y={x}^{2}$ is above the line $y=4$ when $x$ is less than $-2$ and when $x$ is greater than 2.

The points of intersection must be included in the solution, because the inequality has a “greater than or equal to” sign. The answer is thus $x\le -2$ and $x\ge 2$.