e only value that works is the one in the subinterval -4 < x < 4 or (-4, 4). We now need to chec

Step 1: The given inequality is 7 < x < 8.

E only value that works is the one in the subinterval -4

e only value that works is the one in the subinterval -4 < x < 4 or (-4, 4). We now need to check whether the endpoi equality $|x| < 4$. Since the inequality is strict, the endpoints do not satisfy the inequality equation. Therefore, the solution -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 Practice State your answer in interval notation. • You can use "inf" and "-inf" for $\infty$ and $-\infty$. • Use the union operator "U" to describe intervals with separate regions. • If the inequality is true for all real numbers, use "R" to-represent "all real numbers". • If the inequality is never true, use "()" or "EmptySet" (case-sensitive) to represent "no solutions". a. $|x| \le 8$ is true on the interval b. $|x| \le 6$ is true on the interval c. $|x| \le 12$ is true on the interval d. $|x| \le 24$ is true on the interval e. $|x| \le 75$ is true on the interval

Transcript

00:01 Hello everyone so according to the given problem we have given these five inequalities and we need to find the interval so let's consider the first inequality which is a mod of 6x minus 5 is greater than 6 so in order to find the interval we can write this inequality in a square root form so let's take a square root of 6x minus 5 and take a square which is greater than 6 this thing implies that the under under the square root thing will be positive you know so the mod the modulus of any any any variable will give you the positive value right so so we can write this as a plus minus 6x minus 5 greater than 6 so for the positive value for the positive value we write 6x minus 5 greater than 6 then solve for let's take this minus 5 to the right hand side so this is greater than 11 so x is greater than 11 11 11 11 11 by 5 right so one thing is this and the second part let's take the minus sign this is minus 6x minus 5 is greater than 6 now multiply this multiply minus on both hand side so the inequality sign will change so 6x minus 5 less than minus of 6 so this is 6x minus 5 less than now let's take this minus 5 to the right hand side so this will give us minus 1 so x is less than minus 1 by 6 right so yeah now we can we can define the interval so for this the interval of x is a from 11 by 5 to plus infinity means of 11 by 6 from 11 by 6 to infinity and for this the limit for this we can define minus infinity to minus of 1 by 6 so the interval of x for first can be written as x belongs to minus infinity to minus of 1 by 6 union 11 by 6 to infinity okay so this is the answer of first one now for the second one second one is 10 minus of 6x greater than equal to 5 is that it yeah this one okay so for to solve this one we will do the same thing let's take under the square root 10x minus 6x square greater than 5 so this will give you two values one is positive one is negative so positive so for positive one we will write this as a 10 minus 6x greater than equal to 5 and for the negative value we write minus of 10 minus 6x greater than equal to 5 so let's solve this one so this is a let's take 10 to the right hand side so this is minus of 6x greater than equal to this is this will become 5 minus 10 5 minus 10 so that will give you minus of 5 now multiply minus sign both inside so the inequality sign will reverse so 6x will become 6x less than 5 so x is less than 5 by 6 now for this multiply minus sign both inside so 10x minus 6x will be less than equal to 5 right now now do the same thing again so that will give you this is okay so these are basically same 6x minus equal to 5 minus 10 so this will give you the same answer again x is less than equal to 5 by 6 okay so for this value only this inequality holds so the answer is x belongs to minus infinity to 5 by 6 okay there is us there is a slightly mistake here this i didn't consider this minus sign here so let me consider this minus sign that will give us the correct answer okay so there is a minus sign because we are multiplying this this minus here but okay and yeah that's fine so now this is 10 minus 6x less than equal to minus 5 so this is minus 6x less than equal to minus 5 so this is minus 6x 15.

04:48 Now we can multiply with the minus sign both sides so that will give us 6x greater than equal to 15.

04:56 So the value of x is greater than is basically 5 by right.

05:05 So the interval will change.

05:12 Right...

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