2.2.1: Exercises 2.2
- Page ID
- 62900
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Terms and Concepts
Exercise \(\PageIndex{1}\)
In your own words, explain the what is meant by a strict inequality.
- Answer
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A strict inequality means we have \(>\) or \(<\).
Exercise \(\PageIndex{2}\)
In your own words, describe the two ways we can have break points.
- Answer
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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
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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
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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
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\(x \in [-3,10]\)
Exercise \(\PageIndex{6}\)
\(x \geq -5\) and \(x>2\)
- Answer
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\(x \in (2, \infty)\)
Exercise \(\PageIndex{7}\)
\(x \geq -5\) and \(x<2\)
- Answer
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\(x \in [-5,2)\)
Exercise \(\PageIndex{8}\)
\(x \leq -5\) and \(x>2\)
- Answer
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no values of \(x\) satisfy this statement
Exercise \(\PageIndex{9}\)
\(x \geq -5\) or \(x>2\)
- Answer
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\(x \in [-5, \infty)\)
Exercise \(\PageIndex{10}\)
\(x\leq 4\) and \(x>-6\)
- Answer
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\(x \in (-6,4]\)
Exercise \(\PageIndex{11}\)
\(x > 4\) or \(-2>x\)
- Answer
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\(x \in (-\infty,-2)\cup (4,\infty)\)
In exercises \(\PageIndex{12}\) - \(\PageIndex{14}\), write each statement using inequalities.
Exercise \(\PageIndex{12}\)
\(x \in [3,4)\cup (4,\infty)\)
- Answer
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\(3 \leq x <4\) or \(x>4\)
Exercise \(\PageIndex{13}\)
\(x \in [-2,4)\)
- Answer
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\(-2 \leq x < 4\)
Exercise \(\PageIndex{14}\)
\(x \in (5,6] \cup [7,8)\)
- Answer
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\(5 < x \leq 6\) 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
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\(\displaystyle x \in [2,4)\)
Exercise \(\PageIndex{16}\)
\(\displaystyle x^2-2x+ 8 \leq 2x+5\)
- Answer
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\(\displaystyle x \in [1,3]\)
Exercise \(\PageIndex{17}\)
\(\displaystyle x^2+2x >15\)
- Answer
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\(\displaystyle x \in (-\infty,-5) \cup (3,\infty)\)
Exercise \(\PageIndex{18}\)
\(\displaystyle -x^2+7x+10 \geq 0\)
- Answer
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\(\displaystyle x \in \Big[\frac{7-\sqrt{89}}{2}, \frac{7+\sqrt{89}}{2} \Big]\)
Exercise \(\PageIndex{19}\)
\(\displaystyle \frac{x+3}{x-2} -2 \leq 0\)
- Answer
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\(\displaystyle x \in (-\infty,2) \cup [7, \infty)\)
Exercise \(\PageIndex{20}\)
\(\displaystyle 2x^2-4x-45 \leq -4x+5\)
- Answer
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\(\displaystyle x \in [-5,5]\)
Exercise \(\PageIndex{21}\)
\(\displaystyle \frac{3x+1}{x-2} \leq 2\)
- Answer
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\(\displaystyle x \in [-5,2)\)
Exercise \(\PageIndex{22}\)
\(\displaystyle 1+x<7x+5\)
- Answer
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\(x \in (\frac{-2}{3}, \infty)\)
Exercise \(\PageIndex{23}\)
\(\displaystyle \theta^2 - 5\theta \leq -6\)
- Answer
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\(\displaystyle \theta \in [2,3]\)
Exercise \(\PageIndex{24}\)
\(\displaystyle y^3+3y^2 > 4y\)
- Answer
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\(\displaystyle y \in (-4,0) \cup (1,\infty)\)
Exercise \(\PageIndex{25}\)
\(\displaystyle x^3-x^2 \leq 0\)
- Answer
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\(\displaystyle x \in (-\infty,1]\)
Exercise \(\PageIndex{26}\)
\(\displaystyle \frac{x^2+3x+2}{x^2-16} \geq 0\)
- Answer
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\(\displaystyle x \in (-\infty,-4)\cup [-2,-1] \cup (4,\infty)\)