Question

Please give the answer

Answer 1

Answer: See the attached image below for the filled in table.

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Step-by-step explanation:

The domain is the set of all allowed x inputs of a function.

The term "radicand" refers to the stuff under the square root.

We cannot have a negative number under the square root. Therefore, the radicand must be 0 or larger.

The first function must have
`4x+6 \ge 0` which becomes
`4x \ge -6` and leads to
`x \ge -(6)/(4)` aka
`x \ge -(3)/(2)`

So x = -3/2 is the smallest input allowed. Which is where the interval notation of
`\left[-(3)/(2), \infty)` comes from. It's the interval spanning from -3/2 (inclusive) to infinity. This represents the set of all possible x inputs of the first function.

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Follow this same idea for problem 2. The steps would be something like this:

`-20x - 6 \ge 0\\\\-20x \ge 6\\\\x \le (6)/(-20)\\\\x \le -(3)/(10)\\\\`

The inequality sign flips because we divided both sides by a negative number. That then leads to the interval notation of
`(-\infty, -(3)/(10)\big]`

The interval spans from negative infinity (exclusive) to -3/10 (inclusive).

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Problem 3 would have these steps

`5x-16 \ge 0\\\\5x\ge 16\\\\x\ge (16)/(5)\\\\\big[(16)/(5), \infty)\\\\`

No sign flip happens since we divided both sides by a positive number.

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Lastly, here are the steps for problem 4

`20+6x \ge 0\\\\6x \ge -20\\\\x \ge -(20)/(6)\\\\x \ge -(10)/(3)\\\\\big[-(10)/(3), \infty)`

Don't forget to use the square bracket to include the endpoints mentioned. We always use curved parenthesis for either infinity because we can't ever reach infinity. In a sense, we "approach" infinity.

Once again, all of this is summarized in the filled out table shown below (attached image). Feel free to ask any questions if they come to mind.