Answer the following questions based on the fundamental properties of inequalities.

MATHEMATICS · Multi-part · CBSE · Mathematics · Linear Inequalities · Linear Inequalities

Problem

Answer the following questions based on the fundamental properties of inequalities. (i) Prove that if

a<b

and

c<0

, then

\dfrac a c>\dfrac b c

. Give a logical justification for the reversal of the inequality sign. (ii) Show that adding the same real number

k

to both sides of an inequality does not alter its direction. Illustrate your reasoning with a general proof for

a<b \Rightarrow a+k<b+k

. (iii) Using the results of parts (i) and (ii), transform the inequality

-4x+3>11

step by step until

x

is isolated, and state the final solution set for

x