3.5: Lin. Inequal. in One and Two Var.- Answers to the Homework Exercises
- Page ID
- 45042
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Linear Inequalities
- \((-5,\infty )\)
- \((-\infty ,5]\)
- \((-\infty ,1]\)
- \(x<2;\: (-\infty ,2)\)
- \(x\geq 5;\: [5,\infty )\)
- \(x>-2;\: (-2,\infty )\)
- \([110,\infty )\)
- \([-6,\infty )\)
- \([11,\infty )\)
- \((-\infty ,-1)\)
- \((8,\infty )\)
- \((-\infty ,0)\)
- \((-\infty ,\infty )\)
- \((-\infty ,-6]\)
- \([1,\infty )\)
- \((-\infty ,-10]\)
- \((-\infty ,5]\)
- \([0,\infty )\)
- \((1,\infty )\)
- \([-6,-2]\)
- \((-2,2)\)
- no solution
- \([-2,4]\)
Compound Inequalities
- \(n ≤ −9\) or \(n ≥ 2\): \((−∞, −9] ∪ [2, ∞)\)
- \(x < −7:\: (−∞, −7)\)
- \(b < 5:\: (−∞, 5)\)
- \(−6 ≤ x ≤ −2 :\: [−6, −2]\)
- \(−2 < x < 2 :\: (−2, 2)\)
- No solution; \(∅\)
- \(5 ≤ x < 19 :\: [5, 19)\)
- \(k ≥ 2\) or \(k < −20:\: ( − ∞, −20) ∪ [2, ∞)\)
- \(m ≥ −4\) or \(m < −5 :\: (−∞, −5) ∪ [−4, ∞)\)
- \(n < −7\) or \(n > 8:\: ( − ∞ − 7), ∪(8, ∞)\)
- \(−2 ≤ n ≤ 6 :\: [−2, 6]\)
- \(−9 ≤ x ≤ 0 :\: [−9, 0]\)
- No solution; \(∅\)
- \(x ≤ 0\) or \(x > 8:\: ( − ∞, 0] ∪ (8, ∞)\)
- \(n < −14\) or \(n ≥ 17:\: ( − ∞, −14) ∪ [17, ∞)\)
- all real numbers
Absolute Value Inequalities
- \(-3,\: 3\)
- \(-4,\:8\)
- \(-\frac{7}{3},\:\frac{11}{3}\)
- \(1,\:4\)
- \((−∞, −6) ∪ (0, ∞)\)
- \((−∞, −1] ∪ [3, ∞)\)
- \([1, 3]\)
- \(\left[ 1,\frac{3}{2}\right]\)
- \([2, 4]\)
- \(-8,\: 8\)
- \(-4,\: 20\)
- \(-7,\: 2\)
- \((−∞, 5) ∪ (5, ∞)\)
- \((−∞, −1) ∪ (5, ∞)\)
- \(\left[ -\frac{4}{3},2\right]\)
- \(\left[\frac{1}{2},1\right]\)
- \(\left[-2 ,-\frac{4}{3}\right]\)
- \([-3,-2]\)
Linear Inequalities in Two Variables
- yes; no
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