12.3: Exercises
- Page ID
- 49026
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Solve for \(x\).
- \(5x+6\leq 21\)
- \(3+4x> 10x\)
- \(2x+8\geq 6x+24\)
- \(9-3x< 2x-13\)
- \(-5 \leq 2x+5\leq 19\)
- \(15> 7-2x\geq 1\)
- \(3x+4 \leq 6x-2\leq 8x+5\)
- \(5x+2< 4x-18\leq 7x+11\)
- Answer
-
- \(x \leq 3\)
- \(\dfrac{1}{2}>x\)
- \(-4 \geq x\)
- \(x>\dfrac{22}{5}\)
- \(-5 \leq x \leq 7\)
- \(-4<x \leq 3\)
- \(x \geq 2\) (this then also implies \(x \geq-\dfrac{7}{2}\))
- no solution
Solve for \(x\).
- \(x^2-5x-14>0\)
- \(x^2-2x\geq 35\)
- \(x^2-4\leq 0\)
- \(x^2+3x-3<0\)
- \(2x^2+2x\leq 12\)
- \(3x^2<2x+1\)
- \(x^2-4x+4>0\)
- \(x^3-2x^2-5x+6\geq 0\)
- \(x^3+4x^2+3x+12 <0\)
- \(-x^3-4x<-4x^2\)
- \(x^4-10x^2+9\leq 0\)
- \(x^4-5x^3+5x^2+5x<6\)
- \(x^4-5x^3+6x^2>0\)
- \(x^5-6x^4+x^3+24x^2-20x\leq 0\)
- \(x^5-15x^4+85x^3-225x^2+274x-120\geq 0\)
- \(x^{11}-x^{10}+x-1\leq 0\)
- Answer
-
- \((-\infty,-2) \cup(7, \infty)\)
- \((-\infty,-5] \cup[7, \infty)\)
- \([-2,2]\)
- \(\left(\dfrac{-3-\sqrt{21}}{2}, \dfrac{-3+\sqrt{21}}{2}\right)\)
- \([-3,2]\)
- \(\left(-\dfrac{1}{3}, 1\right)\)
- \(\mathbb{R}-\{2\}\)
- \([-2,1] \cup[3, \infty)\)
- \((-\infty,-4)\)
- \((0,2) \cup(2, \infty)\)
- \([-3,-1] \cup[1,3]\)
- \((-1,1) \cup(2,3)\)
- \((-\infty, 0) \cup(0,2) \cup(3, \infty)\)
- \((-\infty,-2] \cup[0,1] \cup[2,5]\)
- \([1,2] \cup[3,4] \cup [5, \infty)\)
- \((-\infty, 1]\)
Find the domain of the functions below.
- \(f(x)=\sqrt{x^2-8x+15}\)
- \(f(x)=\sqrt{9x-x^3}\)
- \(f(x)=\sqrt{(x-1)(4-x)}\)
- \(f(x)=\sqrt{(x-2)(x-5)(x-6)}\)
- \(f(x)=\dfrac{5}{\sqrt{6-2x}}\)
- \(f(x)=\dfrac{1}{\sqrt{x^2-6x-7}}\)
- Answer
-
- \(D=(-\infty, 3] \cup[5, \infty)\)
- \(D=(-\infty,-3] \cup[0,3]\)
- \(D=[1,4]\)
- \(D=[2,5] \cup[6, \infty)\)
- \(D=(-\infty, 3)\)
- \(D=(-\infty,-1) \cup(7, \infty)\)
Solve for \(x\).
- \(\dfrac{x-5}{2-x}>0\)
- \(\dfrac{4x-4}{x^2-4}\geq 0\)
- \(\dfrac{x-2}{x^2-4x-5}< 0\)
- \(\dfrac{x^2-9}{x^2-4}\geq 0\)
- \(\dfrac{x-3}{x+3}\leq 4\)
- \(\dfrac{1}{x+10}> 5\)
- \(\dfrac{2}{x-2}\leq \dfrac{5}{x+1}\)
- \(\dfrac{x^2}{x+4}\leq x\)
- Answer
-
- \((2,5)\)
- \((-2,1] \cup(2, \infty)\)
- \((-\infty,-1) \cup(2,5)\)
- \((-\infty,-3] \cup (-2,2) \cup[3, \infty)\)
- \((-\infty,-5] \cup(-3, \infty)\)
- \((-10,-9.8)\)
- \((-1,2) \cup [4, \infty)\)
- \((-\infty,-4) \cup[0, \infty)\)
Solve for \(x\).
- \(|2x+7|>9\)
- \(|6x+2|<3\)
- \(|5-3x|\geq 4\)
- \(|-x-7|\leq 5\)
- \(|1-8x|\geq 3\)
- \(1>\left|2+\dfrac{x}{5}\right|\)
- Answer
-
- \((-\infty,-8) \cup(1, \infty)\)
- \(\left(\dfrac{-5}{6}, \dfrac{1}{6}\right)\)
- \(\left(-\infty, \dfrac{1}{3}\right] \cup[3, \infty)\)
- \([-12,-2]\)
- \(\left(-\infty,-\dfrac{1}{4}\right] \cup\left[\dfrac{1}{2}, \infty\right)\)
- \((-15,-5)\)