In Exercises 1–26, graph each inequality. y≥log_2(x+1)

Question

Accepted Answer

Welcome back. I am so glad you're here. We are given this inequality we have why is greater than, or equal to the log of X -3. And we are asked to graph that inequality. Well, first, I'm going to bring this down where we've got a little bit more room. And next, I'm going to draw a rough sketch of a coordinate plane. So we can draw our graph. We'll have a vertical Y axis and a horizontal X axis. They come together at the origin in the middle. Now to graph an inequality, we first need to graph the equality. So we have to set it as y equals the log of X -3. And because our inequality is greater than or equal to it will include our equality. So when we draw the line for the equality, that's part of what we're graphing. Now, in order to graph this function Y equals the log of X minus three, we need to think what is our base function here? And the base function is a five X equals the log of X. And so we recall from previous lessons that for the F of X equals the log of X, we are going to start from negative infinity and there is an ASM tote along the Y axis at X equals zero. So it's going to come up along the Y axis getting very close but not touching it. And then it's going to have an X intercept intercept at 10. And then it's going to curve out like this heading toward positive infinity. So we've got 10 for our X intercept and we have a vertical ascent tote at Y or at X equals zero, excuse me at X equals zero. And so that's our affects equals the log of X. Now, what's happening in our Y equals log of X minus three. Well, the difference is this minus three here, The -3 is inside of the parentheses. It's happening directly to that X. So that indicates a horizontal shift and because it is subtracting three, it is a horizontal shift to the right three units. So two things will change here. Our ASM tote along the Y axis is going to shift to the right three units. So that's going to be along X equals three. There's the ASM tote and it will again come up along the ascent to being very close to it and then it will intercept the X axis at three to the right of 10, which will be 40. So we've got an X intercept at 40 and there is our Y. Equals The Log of X -3. So we've got that graft next, we need to look back and say, all right, where is it true that why is greater than the log of X -3? And in order to do that, we choose test points, we choose a test point up here above our function line and we choose a test point down below our function line. And it has to be within our domain. It's going to be to the right of that ASAM tote of X equals three. And so I'm going to choose, you can choose anything as long as it's within that domain and range. I am going to choose the 30. 20 for a test point to test above the function line. And I'm going to choose the 200.0.13 negative two to choose the test point below our function line. And we're going to see where this is true when we plug in the X and Y values. So let's start with 13 20 our Y value there's 20. So we're putting in 20 is greater than or equal to the log of parentheses. The X value is 13. So 13 minus three, Simplifying What's in the Parentheses, -3 is 10. So we have the log of 10. Will we recall from previous lessons that when there's no base written for our log, that's log base 10 of 10? And the log base 10 of 10 simplifies to one that's why I chose 13 for the X value? So is it true that 20 is greater than or equal to one? Yes, it is. So this part gets shaded in up here above our function line. Now we can test the other test 0.2, 13, negative two, we put negative two and for Y is greater than or equal to the log of still 13 minus three. And we know that the right hand side simplifies to one. So is it true that negative two is greater than, or equal to one? No, it's not. So we know that that test point doesn't work and we know that it's only true for the values above the function line and including the function line as well. So we have got this graft. Now, let's bring this up and let's see which answer choice matches. Okay. So we're looking for a function line that intercepts the X axis at 40 has an assam tote along the line X equals three and is shading above our function line. Well, answer choices C and D have the ASM total long X equals three and they have 40 as the X intercept. But answer choice D has shaded in the space above our function lines. So answer choice D is the correct answer. Well done. We'll catch you on the next one.

In Exercises 1–26, graph each inequality. y≥log_2(x+1)

Key Concepts

Inequalities

Logarithmic Functions

Graphing Techniques

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