2.2.1: Exercises 2.2

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Terms and Concepts

Exercise \(\PageIndex{1}\)

In your own words, explain the what is meant by a strict inequality.

Answer: A strict inequality means we have \(>\) or \(<\).

Exercise \(\PageIndex{2}\)

In your own words, describe the two ways we can have break points.

Answer: We have break points when the equality statement is true or where the statement is undefined.

Exercise \(\PageIndex{3}\)

Does a statement always switch from true to false at a break point? Give an example to support your argument.

Answer: No, the statement \(x^2 >0\) is always true, but has a break point at \(x=0\).

Exercise \(\PageIndex{4}\)

What methods can you use to find the break points of a quadratic equality?

Answer: We need to move everything to one side, and then we can factor or use the quadratic formula to find the roots.

Problems

In exercises \(\PageIndex{5}\) - \(\PageIndex{11}\), write each statement in simplified interval notation.

Exercise \(\PageIndex{5}\)

\(-3\leq x \leq 10\)

Answer: \(x \in [-3,10]\)

Exercise \(\PageIndex{6}\)

\(x \geq -5\) and \(x>2\)

Answer: \(x \in (2, \infty)\)

Exercise \(\PageIndex{7}\)

\(x \geq -5\) and \(x<2\)

Answer: \(x \in [-5,2)\)

Exercise \(\PageIndex{8}\)

\(x \leq -5\) and \(x>2\)

Answer: no values of \(x\) satisfy this statement

Exercise \(\PageIndex{9}\)

\(x \geq -5\) or \(x>2\)

Answer: \(x \in [-5, \infty)\)

Exercise \(\PageIndex{10}\)

\(x\leq 4\) and \(x>-6\)

Answer: \(x \in (-6,4]\)

Exercise \(\PageIndex{11}\)

\(x > 4\) or \(-2>x\)

Answer: \(x \in (-\infty,-2)\cup (4,\infty)\)

In exercises \(\PageIndex{12}\) - \(\PageIndex{14}\), write each statement using inequalities.

Exercise \(\PageIndex{12}\)

\(x \in [3,4)\cup (4,\infty)\)

Answer: \(3 \leq x <4\) or \(x>4\)

Exercise \(\PageIndex{13}\)

\(x \in [-2,4)\)

Answer: \(-2 \leq x < 4\)

Exercise \(\PageIndex{14}\)

\(x \in (5,6] \cup [7,8)\)

Answer: \(5 < x \leq 6\) or \(7 \leq x <8\)

In exercises \(\PageIndex{15}\) - \(\PageIndex{26}\), solve the given inequality and express your answer in interval notation.

Exercise \(\PageIndex{15}\)

\(\displaystyle \frac{x-2}{x-4} \leq 0\)

Answer: \(\displaystyle x \in [2,4)\)

Exercise \(\PageIndex{16}\)

\(\displaystyle x^2-2x+ 8 \leq 2x+5\)

Answer: \(\displaystyle x \in [1,3]\)

Exercise \(\PageIndex{17}\)

\(\displaystyle x^2+2x >15\)

Answer: \(\displaystyle x \in (-\infty,-5) \cup (3,\infty)\)

Exercise \(\PageIndex{18}\)

\(\displaystyle -x^2+7x+10 \geq 0\)

Answer: \(\displaystyle x \in \Big[\frac{7-\sqrt{89}}{2}, \frac{7+\sqrt{89}}{2} \Big]\)

Exercise \(\PageIndex{19}\)

\(\displaystyle \frac{x+3}{x-2} -2 \leq 0\)

Answer: \(\displaystyle x \in (-\infty,2) \cup [7, \infty)\)

Exercise \(\PageIndex{20}\)

\(\displaystyle 2x^2-4x-45 \leq -4x+5\)

Answer: \(\displaystyle x \in [-5,5]\)

Exercise \(\PageIndex{21}\)

\(\displaystyle \frac{3x+1}{x-2} \leq 2\)

Answer: \(\displaystyle x \in [-5,2)\)

Exercise \(\PageIndex{22}\)

\(\displaystyle 1+x<7x+5\)

Answer: \(x \in (\frac{-2}{3}, \infty)\)

Exercise \(\PageIndex{23}\)

\(\displaystyle \theta^2 - 5\theta \leq -6\)

Answer: \(\displaystyle \theta \in [2,3]\)

Exercise \(\PageIndex{24}\)

\(\displaystyle y^3+3y^2 > 4y\)

Answer: \(\displaystyle y \in (-4,0) \cup (1,\infty)\)

Exercise \(\PageIndex{25}\)

\(\displaystyle x^3-x^2 \leq 0\)

Answer: \(\displaystyle x \in (-\infty,1]\)

Exercise \(\PageIndex{26}\)

\(\displaystyle \frac{x^2+3x+2}{x^2-16} \geq 0\)

Answer: \(\displaystyle x \in (-\infty,-4)\cup [-2,-1] \cup (4,\infty)\)