Video: Solving Radical Inequalities in One Variable Involving Factorisation

Find algebraically the solution set of the inequality √(𝑥² + 16𝑥 + 64) < 19.

04:59

Video Transcript

Find algebraically the solution set of the inequality square root of 𝑥 squared plus 16𝑥 plus 64 is less than 19.

So, if we take a look at this problem, the first thing we want to do is square each side of the inequality. And when we do that, we’re gonna have 𝑥 squared plus 16𝑥 plus 64 is less than 361. And that’s because 19 squared is 361. And then, there’re couple of methods now to solve this.

So, first of all, we can take a look at 𝑥 squared plus 16𝑥 plus 64 and factor it. And when we do that, we’ll get 𝑥 plus eight all squared is less than 361. And we can see that because the 𝑥 squared plus 16𝑥 plus 64. If we multiply eight by eight, we get 64. And if we add eight and eight, we get 16. So therefore, it is the factor of 𝑥 squared plus 16𝑥 plus 64 to square 𝑥 plus eight.

So, now to enable us to find our two critical values, we’re gonna set it equal to 361. So, we’ve got 𝑥 plus eight all squared equals 361. And this is because we want to find the solutions for 𝑥, which will be our two points on our graph. So then, if we square root both sides of the equation, we’re gonna get 𝑥 plus eight is equal to positive or negative 19.

So therefore, if we subtract eight from each side of the equation, we’re gonna get 𝑥 is equal to 19 minus eight or negative 19 minus eight. So therefore, our critical values are 11 or negative 27. Okay, so, that’s how we found the two values that we want to look at on our graph in a second using this particular method. I said I’ll show you another method. And the other method would be by completing the square.

So, if we’re gonna solve it by completing the square, first of all, we could subtract 361 from each side. So, we get 𝑥 squared plus 16𝑥 minus 297 is less than zero. And then, we could complete the square. So, we divide the 𝑥-coefficient by two and have that inside the parentheses. So, we’ve got 𝑥 plus eight all squared then minus eight squared then minus 297. So then, we arrive at the same point that we did in the other method. So, we get 𝑥 plus eight all squared minus 361 is less than zero.

So then, we’d set this equal to zero to find our critical points. So, we get 𝑥 plus eight all squared minus 361 is equal to zero. And then, we’d solve in the same way as the left-hand side. And we get our two values for 𝑥, which are 𝑥 is equal to 11 or 𝑥 is equal to negative 27 as our solutions if we make it equal to zero.

So, this is all well and good, however what we want to find is the solution set of the inequality. Whenever you solve a problem like this and you need to show the solution set of an inequality, then we need to show where our answer comes from. So, I’m gonna do that using a quick sketch.

So, if we draw a quick sketch of our graph, well, we know it’s gonna be a U-shaped parabola because it’s 𝑥 squared. So, we’ve got 𝑦 is equal to 𝑥 squared plus 16𝑥 minus 297. I’ve just chosen to write it in this way because this is where we’d have it if we were gonna make it equal to zero. And then, we can see that it crosses the 𝑥-axis at negative 27 and 11. And this is because these are the two values that we found for the solutions.

Well, then, if we check out our inequality, we’d see that we want to find out where this is less than zero, so where 𝑥 squared plus 16𝑥 minus 297 is less than zero. So therefore, we’re interested in this region here which is below the 𝑥-axis. So, we can see that our solution’s gonna lie between negative 27 and 11. So therefore, we can say that the inequality that would satisfy this problem is 𝑥 is greater than negative 27 but less than 11.

However, we want it written as a solution set. So therefore, to write it as a solution set, we write it like this. And we’ve got the outward-facing brackets because it means that it’s not including the values negative 27 and 11. So therefore, we found the solution set of the inequality the square root of 𝑥 squared plus 16𝑥 plus 64 is less than 19. Because it gives all the values from negative 27 to 11 but not including negative 27 or 11.

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