Math Lesson 10.3.3 - The Standard Form of a Linear Inequality from a Given Graph

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Welcome to our Math lesson on The Standard Form of a Linear Inequality from a Given Graph, this is the third lesson of our suite of math lessons covering the topic of Graphing Inequalities, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Finding the Standard Form of a Linear Inequality from a Given Graph

Sometimes, we have the graph of a linear inequality given but not the inequality shown by that graph. To find the standard form of that inequality, we must first find the corresponding linear equation representing the boundary line of the inequality. For this, we need the coordinates of two known points A and B of the graph to calculate the gradient k by applying the formula

k = ∆y/∆x = y_B - y_A/x_B - x_A

Then, using the equation of the line

y = kx + n

we can find the constant n by substituting the coordinates of any from the known points.

Finally, looking at the highlighted region on the graph, you can determine the standard form of the given inequality.

Example 3

What inequality is shown in the graph below?

$Math Tutorials: Graphing Inequalities Example$

Solution 3

Let's consider two known points A(0, 1) and B(1, 3) as shown in the figure.

$Math Tutorials: Graphing Inequalities Example$

The gradient k is

k = ∆y/∆x
= y_B - y_A/x_B - x_A
= 3 - 1/1 - 0
= 2/1
= 2

Hence, in the line

y = 2x + n

We substitute for example the coordinates of point A (x = 0 and y = 1) in the above equation. Thus, we obtain

1 = 2 ∙ 0 + n
n = 1

Therefore, the boundary line (which is included in the solution set of our inequality) has the equation

y = 2x + 1

Since we have the region above the graph highlighted, the inequality shown by this graph is

y ≥ 2x + 1

More Graphing Inequalities Lessons and Learning Resources

Inequalities Learning Material
Tutorial ID	Math Tutorial Title	Revision Notes	Revision Questions
10.3	Graphing Inequalities
Lesson ID	Math Lesson Title	Lesson	Video Lesson
10.3.1	Graphing First Order Inequalities with One Variable
10.3.2	Graphing First Order Inequalities with Two Variables
10.3.3	The Standard Form of a Linear Inequality from a Given Graph
10.3.4	Graphing Second-Order Inequalities

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Continuing learning inequalities - read our next math tutorial: Systems of Inequalities

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