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Properties of Inequalities

Lesson · Beginner

Algebra: Inequalities

Basic Properties of Inequalities

a,b,c∈Ra,b,c\in\mathbb{R}
a<b⟹a+c<b+ca<b\Longrightarrow a+c<b+c
a<b⟹a−c<b−ca<b\Longrightarrow a-c<b-c
c>0, a<b⟹ac<bc,ac<bcc>0,\ a<b\Longrightarrow ac<bc,\qquad \frac{a}{c}<\frac{b}{c}
c<0, a<b⟹ac>bc,ac>bcc<0,\ a<b\Longrightarrow ac>bc,\qquad \frac{a}{c}>\frac{b}{c}

Powers of Inequalities

n∈Z+∪{0}, a<b⟹a2n+1<b2n+1n\in\mathbb{Z}^+\cup\{0\},\ a<b\Longrightarrow a^{2n+1}<b^{2n+1}
0≤a<b⟹a2n<b2n0\le a<b\Longrightarrow a^{2n}<b^{2n}
a<b≤0⟹a2n>b2na<b\le0\Longrightarrow a^{2n}>b^{2n}
a2n<b2n⟺∣a∣<∣b∣a^{2n}<b^{2n}\Longleftrightarrow |a|<|b|
ab>0, a<b⟹1a>1bab>0,\ a<b\Longrightarrow \frac1a>\frac1b
ab<0, a<b⟹1a<1bab<0,\ a<b\Longrightarrow \frac1a<\frac1b
0≤a<b⟹a2n<b2n0\le a<b\Longrightarrow \sqrt[2n]{a}<\sqrt[2n]{b}
a<b⟹a2n+1<b2n+1a<b\Longrightarrow \sqrt[2n+1]{a}<\sqrt[2n+1]{b}
bd>0, ab<cd⟹ad<bcbd>0,\ \frac ab<\frac cd\Longrightarrow ad<bc
bd<0, ab<cd⟹ad>bcbd<0,\ \frac ab<\frac cd\Longrightarrow ad>bc
0≤a≤b, 0≤c≤d⟹ac≤bd0\le a\le b,\ 0\le c\le d\Longrightarrow ac\le bd
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