WRITE INEQUALITIES FROM GRAPHS

Inequalities in One Variable from Graphs

In the graph of an inequality in one variable, if there is empty circle, we have to use < and > in the inequality. 

In the graph of an inequality in one variable, if there is filled circle, we have to use ≤ and ≥ in the inequality. 

More clearly, 

Example 1 :

Write the inequality for the graph given below.

Solution :

In the above graph, we find the filled circle. So we have to use the sign ≤ or ≥.

Now we have to look into the shaded portion. Since the shaded region is in left hand side from the filled circle, we have to use the sign "≤ ".

The inequality for the above graph is x ≤ 4.

Example 2 :

Write the inequality for the graph given below.

Solution :

In the above graph, we find the unfilled circle. So we have to use the sign < or >.

Now we have to look into the shaded portion. Since the shaded region is in right hand side from the unfilled circle, we have to use the sign "> ".

The inequality for the above graph is x > -6. 

Example 3 :

Write the inequality for the graph given below.

Solution :

In the above graph, we find the unfilled circle. So we have to use the sign < or >.

Now we have to look into the shaded portion. Since the shaded region is in left hand side from the unfilled circle, we have to use the sign "<".

The inequality for the above graph is x  < 1.

Example 4 :

Write the inequality for the graph given below.

Solution :

In the above graph, we find the unfilled circle. So we have to use the sign ≤ or ≥.

Now we have to look into the shaded portion. Since the shaded region is in right hand side from the unfilled circle, we have to use the sign "≥".

The inequality for the above graph is x ≥ 1.

Inequalities in Two Variables from Graph

To find linear inequalities in two variables from graph, first we have to find two information from the graph. 

(i) Slope 

(ii) y -intercept

By using the above two information we can easily get a linear linear equation in the form y = mx + b.

Here "m" stands for slope and "b" stands for y-intercept.

Now we have to notice whether the given line is solid line or dotted line.

• If the graph contains the dotted line, then we have to use one of the signs < or >.
• If the graph contains the solid line, then we have to use one of the signs ≤  or ≥. 

Example 5 :

Write the inequality for the graph given below.

Solution :

From the above graph, first let us find the slope and y-intercept.

Rise  =  -3 and Run  =  1

Slope  =  -3 / 1  =  -3

y-intercept  =  4

So, the equation of the given line is

y  =  -3x + 4

But we need to use inequality which satisfies the shaded region.

Since the graph contains solid line, we have to use one of the signs ≤  or ≥.

To fix the correct sign, let us take a point from the shaded region.

Take the point (2, 1) and apply it in the equation

y  =  -3x + 4

1  =  -3(2) + 4

1  =  -6 + 4

1  =  - 2

Here 1 is greater than -2, so we have to choose the sign ≥ instead of equal sign in the equation y = -3x + 4.

Hence, the required inequality is

y ≥ -3x + 4

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