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Writing a Linear Inequality from a Graph

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Today we're going to talk about writing a linear inequality from a graph. Remember, a linear inequality is similar to a linear equation, except for we use an inequality symbol instead of an equal sign. So we're going to do some examples of how to interpret the graph of an inequality and use that to write the linear inequality that it represents.

So for my first example, I've got the graph of a linear inequality. And I want to write an equation for this in slope-intercept form. So I'm going to do that in the same way that I would write an equation in slope-intercept form by identifying the slope and then identifying the y-intercept.

So I see that two points for this linear inequality are 5, 0 and 0, negative 5. So I can use these two points to calculate my slope. So my slope is going to be equal to the difference in my y values-- 0 minus negative 5-- over the difference in my x values-- 5 minus 0. So simplifying this, 0 minus negative 5 is just going to give me 5. And 5 minus 0 is also going to give me 5. 5 divided by 5 is 1. So the slope of this inequality is 1.

Then I can see that where my inequality intercepts my y-axis is at negative 5. So that means that my y-intercept is going to be negative 5. So I can use my slope and my y-intercept to begin to write my inequality. So I will have y, some inequality symbol-- I don't know yet which one it will be-- my slope is 1, and my y-intercept was negative 5.

So now that I have my slope and my y-intercept, I can determine what my inequality symbol is going to be so first I notice that I have a dotted line here instead of a solid line. So that means that my symbol is either going to be strictly less than or strictly greater than, not less than or equal to or greater than or equal to.

Then I notice that my shaded region is below this line, the dotted line. So that tells me that my inequality symbol is going to be less than because it's below the line, whereas if the shaded region was above the dotted line that would be greater than. So my inequality is going to be a less than symbol which makes the inequality for this graph y is less than 1x minus 5.

So for my second example, I've got another graph of a linear inequality. And I want to represent this graph with its linear inequality. So I know that the equation of a graph of a perfectly vertical line can be represented as x is equal to whatever value that line crosses on the x-axis. So if this were just an equation we could represent this line as x is equal to negative 4.

But because we have a shaded region, we know that we're going to represent it with an inequality. And when we determine which inequality symbol we want to use, we need to think about the fact that if we have a shaded region that is to the left of our value, we're going to be using either a less than or a less than or equal to inequality symbol. And if we have a shaded region to the right of our vertical line, then we're going to be using a greater than or greater than or equal to inequality symbol.

So because our shaded region is to the left, we know that we're going to have either x is less than negative 4 or x is less than or equal to negative 4. And because I have a solid line, I know that it's going to be less than or equal to negative 4. So the graph of this inequality can be represented with x is less than or equal to negative 4.

So for my last example, I again have the graph of an inequality, and I want to write the inequality that's associated with this graph. So I see that I have a horizontal line. And a horizontal line can be represented with an equation y is equal to whatever value it passes through on the y-axis. So if this were just a graph of an equation instead of an inequality, this line would be represented with y is equal to 3.

However, since we have an inequality, we need to decide which kind of inequality symbol we need to use. So first I can see that, because it's a dotted line, it's just going to be either strictly less than 3 or strictly greater than 3. And then I notice that my shaded region is above the 3, which is going to tell me that I'm going to use y is strictly greater than 3. If my shaded region were below the 3, then I would be using y is less than 3. So again, because the shaded region is above my dotted line, I'm going to have y is strictly greater than-- again, because it's a dotted line, it's not greater than or equal to-- and the value of 3.

So let's go over our key points from today. As usual, make sure you have them in your notes if you don't already so you can refer to them later.

When graphing a linear inequality, the symbols less than and greater than indicate using a dashed line. And the symbols less than or equal to and greater than or equal to indicate using a solid line. When determining the region to shade on the graph, the symbols less than and less than or equal to indicate shading below the line.

And the symbols greater than and greater than or equal to indicate shading above the line. A vertical line is represented with an equation x equals a, where a is the value on the x-axis that the line crosses. And a horizontal line is represented with the equation y equals a, where a is the value on the y-axis that the line crosses.

So I hope that these key points and examples helped you understand a little bit more about writing linear inequalities from a graph. Keep using your notes and keep on practicing. And soon you'll be a pro. Thanks for watching.