2.3E Exercises
- Page ID
- 152902
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Translate \( (-8, 2] \cup [3, 5) \) into English.
- Translate \( [3, \infty) \) into English.
- Translate "the set of all real numbers except 7" into math.
- Translate "the set of all real numbers strictly between 1 and 4" into math.
- Translate \( \{ x \: | \: x > 0 \} \) into English.
- Translate \( \{ x \: | \: x \neq 0 \} \) into English.
- Draw a number line representing the set \( (-\infty, 5] \).
- What set does the number line below represent?
- Answer
-
- "The set of real numbers strictly greater than \(-8\) but less than or equal to \(2\), together with the set of real numbers greater than or equal to \(3\) but strictly less than \(5\)."
- "The set of all real numbers greater than or equal to \(3\)."
- \( \{ x \: | \: x \neq 7 \} \), or \( x \in \mathbb{R}, x \neq 7 \), etc.
- \( \{ x \: | \: 1 < x < 4 \}\), or in interval notation, \( (1, 4) \).
- "The set of positive real numbers."
- "All real numbers \(x\) except for \(x = 0\)."
- \( \{ x \: | \: -4 < x < 3 \} \) or in interval notation, \( (-4, 3) \).
What is the domain of the expression?
- \( \dfrac{1}{3-x} \)
- \( \dfrac{2}{(x-1)(x-2)} \)
- \( \dfrac{3}{x^2 - 2x - 3} \)
- \( \dfrac{1}{x^2 + 5x + 6} \)
- \( 1+ x+ x^2 + x^3 \)
- \( \sqrt{x+4} \)
- \( \dfrac{1}{\sqrt{x} }\)
- \( (x-7)(x+2) \)
- \( \sqrt[3]{x+8} \)
- Answer
-
- \( \{x \: | \: x \neq 3 \} \)
- \( \{ x \: | \: x \neq 1, x \neq 2 \} \)
- \( \{ x \: | \: x \neq -1, 3 \} \)
- \( \{ x \: | \: x \neq -2, -3 \} \)
- No legality issues, domain is all real numbers.
- \( \{ x \: | \: x \geq -4 \} \)
- \( \{ x \: | \: x > 0 \} \)
- No legality issues, domain is all real numbers.
- Careful! No legality issues because we are allowed to put negative numbers into odd roots like cube roots! All real numbers.
Simplify.
1. \( \dfrac{ 8y}{y^2 - 16} - \dfrac{4}{y-4} \)
2. \( \dfrac{3}{x-3} + \dfrac{2}{x-2} \)
3. \( \dfrac{x^2 - 6x}{x^2 - 1} - \dfrac{3x+2}{1-x^2} \)
4. \( \dfrac{5}{2x - 10} - \dfrac{7}{4x-12} \)
- Answer
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Hint: What does "simplify" mean in this type of problem? When you see rational expressions added or subtracted, your aim is to get them combined into a single fractional term (factor first to make life easier), expand and simplify the numerator as much as possible, re-factor everything you can, and see if any common factors cancel. Only then are you finished.
1. \( \dfrac{4}{y+4} \)
2. \( \dfrac{5x-12}{(x-3)(x-2)} \)
3. \( \dfrac{x-2}{x+1} \)
4. \( \dfrac{3x+5}{4(x-5)(x-3)} \)
Simplify.
1. \( \dfrac{ 3x}{x^2 -5x + 6} \cdot \dfrac{ 9x -27}{x^3} \)
2. \( \dfrac{2x+1}{x^2 -4} \cdot \dfrac{ x^3 - 4x}{2x^2 -5x - 3} \)
3. \( \dfrac{t^2 + 2t - 3}{t^2 - 2t - 3} \cdot \dfrac{3-t}{3+t} \)
4. \( \dfrac{\frac{x^3}{x+1}}{\frac{x}{x^2 + 2x + 1}} \)
- Answer
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Hint: What does "simplify" mean for this type of problem? Your aim is to factor as much as possible, do any cancelling you can, perform the multiplication or division, and then simplify.
1. \( \dfrac{27}{x^2(x-2)} \)
2. \( \dfrac{x}{x-3} \)
3. \( -\dfrac{t-1}{t+1} \) or \( \dfrac{1-t}{t+1} \)
4. \( x^2(x+1)\)
In the compound fractions, get common denominators and add or subtract as needed in numerator and denominator until you have a single fraction of rational expressions. Then compute and simplify if necessary.
1. \( \dfrac{1 + \frac{1}{x}}{\frac{1}{x} - 1} \)
2. \( \dfrac{ 1+\frac{1}{x-2}}{ 1 - \frac{1}{x+3}} \)
- Answer
-
1. \( \dfrac{x+1}{1-x} \)
2. \( \dfrac{(x-1)(x+3)}{(x-2)(x+2)} \)