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Prove the following facts about inequalities. [Hint:\ [[guide:A5dd35d44b#axiom.viii [[guide:A5dd35d44b#axiom.ix [[guide:A5dd35d44b#axiom.x |||Use,]],]],]], and the meanings of [math]\geq[/math] and [math]\leq[/math]. In each problem you will have to consider several cases separately, e.g. [math]a \gt 0[/math] and [math]a = 0[/math].]

• If [math]a \leq b[/math], then [math]a + c \leq b + c[/math].
• If [math]a \geq b[/math], then [math]a + c \geq b + c[/math].
• If [math]a \leq b[/math] and [math]c \geq 0[/math], then [math]ac \leq bc[/math].
• If [math]a \leq b[/math] and [math] c \leq 0[/math], then [math]ac \geq bc[/math].