2.1: Properties of Inequalities
- Page ID
- 35204
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Here are some important properties of inequalities:
If \(a\), \(b\), and \(c\) are real numbers, then:
Transitive Property if \(a < b\) and \(b < c\) then \(a < c\)
Addition Property if \(a < b\) then \(a + c < b + c\)
Subtraction Property if \(a < b\) then \(a − c < b − c\)
Multiplication Property (Multiplying by a positive number) if \(a < b\) and \(c > 0\) then \(ac < bc\)
Multiplication Property (Multiplying by a negative number) if \(a < b\) and \(c < 0\) then \(ac > bc\)
Division Property (Dividing by a positive number) if \(a < b\) and \(c > 0\) then \(\dfrac{a}{c} < \dfrac{b}{c}\)
Division Property (Dividing by a negative number) if \(a < b\) and \(c < 0\) then \(\dfrac{a}{c} > \dfrac{b}{c}\)
Transitive Property
If 3 < 7 and 7 < 14 then...
Solution
3 < 14
Addition Property
If 3 < 7, then add 4 to both sides.
Solution
3 + 4 < 7 + 4
7 < 11
Subtraction Property
If 3 < 7, subtract 6 on both sides
Solution
3 < 7
3 - 6 < 7 - 6
-3 < 1
Multiplication Property (Multiplying by a positive number)
If 3 < 7, multiply both sides by 5.
Solution
3 < 7
3 * 5 < 5 * 7
15 < 35
Multiplication Property (Multiplying by a negative number)
If 3 < 7, multiply both sides by -4.
Solution
3 < 7
3 * -4 ? -4 * 7
-12 ? -28
-12 > -28 The direction of the inequality is changed.
Division Property (Dividing by a positive number)
If 6 < 8, divide both sides by 2.
Solution
6/2 < 8/2
3 < 4
Division Property (Dividing by a negative number)
If 9 < 15, divide both sides by -3.
Solution
9 < 15
9/-3 ? 15/-3
-3 ? -5
-3 > -5 The direction of the inequality is changed.