Answer: See the attached image below for the filled in table.
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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 [tex]4x+6 \ge 0[/tex] which becomes [tex]4x \ge -6[/tex] and leads to [tex]x \ge -\frac{6}{4}[/tex] aka [tex]x \ge -\frac{3}{2}[/tex]
So x = -3/2 is the smallest input allowed. Which is where the interval notation of [tex]\left[-\frac{3}{2}, \infty)[/tex] 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:
[tex]-20x - 6 \ge 0\\\\-20x \ge 6\\\\x \le \frac{6}{-20}\\\\x \le -\frac{3}{10}\\\\[/tex]
The inequality sign flips because we divided both sides by a negative number. That then leads to the interval notation of [tex](-\infty, -\frac{3}{10}\big][/tex]
The interval spans from negative infinity (exclusive) to -3/10 (inclusive).
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Problem 3 would have these steps
[tex]5x-16 \ge 0\\\\5x\ge 16\\\\x\ge \frac{16}{5}\\\\\big[\frac{16}{5}, \infty)\\\\[/tex]
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
[tex]20+6x \ge 0\\\\6x \ge -20\\\\x \ge -\frac{20}{6}\\\\x \ge -\frac{10}{3}\\\\\big[-\frac{10}{3}, \infty)[/tex]
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.
