# Graphing inequalities

**How to graph inequalities**

Graphical representation plays an important role in mathematics as it makes the overall material easy to understand for everyone. Either they are linear or quadratic, equation or inequalities, they can be shown graphically.

**E2.6: Represent inequalities graphically and use this representation to solve simple linear programming problems.**

For a proper representation of inequalities on the graph, the signs play the most significant role. Different signs mean different things which could be understandable in the following ways.

$$<$$ means “is less than”.

$$>$$ means “is greater than”.

$$\leq$$ means “is less than or equal to”.

$$\geq$$ means is “greater than or equal to”.

Now with the help of the examples, we will dive in and learn more about the graphs and their correct representations.

**Worked examples of graphing inequalities**

**Example 1:** On a pair of axes, shade the region which satisfies the inequality $$x\geq5$$.

**Step 1: Change the inequality into equality by changing the sign of $$\geq$$ to $$=$$**

The equation becomes $$x=5$$.

**Step 2: Draw $$x=5$$ on the graph.**

**Step 3: Shade the region that satisfies the inequality.**

The region to the right of $$x=5$$ satisfies the inequality $$x\geq5$$.

**Example 2:** On the same pair of axes, plot the following inequalities and leave the unshaded region which satisfies all of them simultaneously.

$$x+y\leq8$$, $$x>2$$, $$y\geq3$$.

**Step 1: Change the inequality into an equality.**

The equations become $$x+y=8$$, $$x=2$$, $$y=3$$.

**Step 2: Draw $$x+y=8$$ on the graph.**

Take $$x=0$$ to get $$y=8$$ for one point and take $$y=0$$ to get $$x=8$$ to get the second point.

**Step 3: Plot these points on the graph and join them to get the graph of $$x+y=8$$.**

**Step 4: Choose a random point which does not lie on the line and substitute those values of $$x$$ and $$y$$ in inequality and shade the portion which does not satisfy the inequality.**

Let the point be $$(0,0)$$.

substitute the values of $$x$$ and $$y$$ to get, $$0+0\leq8$$.

The inequality holds true for the point. So, the region in which $$ (0,0) $$ lies, satisfies the inequality. Therefore, it should remain unshaded.

**Step 5: Draw the graph of $$x>2$$ and the region satisfies the inequality remains unshaded.**

The region to the right of $$x=2$$ satisfies the inequality $$x>2$$. So, this region remains unshaded.

**Step 6: Draw the graph of $$y\geq3$$ and the region that satisfies the inequality remains unshaded.**

The region above $$y=3$$ satisfies the inequality $$y\geq3$$. So, this region remains unshaded.

**Step 7: Combine all three inequalities to get the solution which satisfies all the inequalities.**

**Example 3:** Shaen is making cupcakes and pies in the kitchen. She makes $$x$$ cupcakes and $$y$$ pies. She makes at least $$4$$ cupcakes and at least $$3$$ pies but no more than $$12$$ cupcakes and pies altogether. Write an inequality for each statement and graph them leaving the region which satisfies the inequalities unshaded and taking the help of graph, give one solution which satisfies all the inequalities.

**Step 1: Make appropriate inequalities from the given conditions.**

$$x\geq4$$, $$y\geq3$$, $$x+y\leq12$$.

**Step 2: Change the inequalities into equations.**

$$x=4$$, $$y=3$$, $$x+y=12$$.

**Step 3: Draw $$x+y=12$$ on graph.**

Take $$x=0$$ to get $$y=12$$ for one point and take $$x=0$$ to get $$y=12$$ to get second point.

**Step 3: Plot these points on the graph and join them to get the graph of $$x+y=12$$.**

**Step 4: Choose a random point which does not lie on the line and substitute those values of $$m$$ and $$n$$ in the inequality and shade the region which does not satisfies the inequality.**

Let the point be $$(0,0)$$

Substitute the values of $$x$$ and $$y$$ to get, $$0+0\leq12$$.

The inequality holds true for the point. So, the region in which $$(0,0)$$ lies satisfies the inequality. Therefore, it should remain unshaded.

**Step 5: Draw the graph of $$x\geq4$$ and the region satisfies the inequality remains unshaded.**

The region to the right of $$x=4$$ satisfies the inequality $$x\geq4$$. So, this region remains unshaded.

**Step 6: Draw the graph of $$y\geq3$$ and the region satisfies the inequality remains unshaded.**

The region above $$y=3$$ satisfies the inequality $$y\geq3$$. So, this region remains unshaded.

**Step 7: Combine the graph of all three inequalities to get the solution which satisfies all inequalities.**

The unshaded region represents the solution to the system of the inequalities.

**Step 10: From the graph choose one point from the unshaded region to get one solution that satisfies all the three inequalities.**

Here, $$(6,4)$$ can be a solution.