Inequalities - Practice Questions & MCQ

Inequalities

Inequalities are the relationship between two expressions that are not equal to one another. Symbols denoting the inequalities are <, >, ≤, ≥, and ≠.

• x < 4, “is read as x less than 4”,  x ≤ 4, “is read as x less than or equal to 4”.
• Similarly x > 4, “is read as x greater than 4” and x ≥ 4, “is read as x greater than or equal to 4”. 

The process of solving inequalities is the same as of equality but instead of the equality symbol inequality symbol is used throughout the process.

A few rules that are different from equality rules

• If we multiply or divide both sides of the inequality by a negative number, then we reverse the inequality (reversing inequality means > gets converted to < and vice versa, and  ≥ gets converted to ≤ and vice versa) (eg 4>3 means -4<-3)
• If we cross multiply a negative quantity in an inequality, then we reverse the inequality (eg 3 > -2 means -3/2 < 1)
• If we cancel the minus sign from both sides of an inequality, then we reverse the inequality. (eg -3 > -4 means 3 < 4)
• As we usually do not know the sign of a variable term like x, (x-2), etc, so we do not cross-multiply them, as we cannot decide if we have to reverse the sign of inequality or not.

We get a range of solutions while solving inequality which satisfies the inequality, 

for e.g.  a > 3 gives us a range of solutions,  means a ? (3, ∞)

Graphically inequalities can be shown as a region belonging to one side of the line or between lines, for example, inequality -3< x ≤ 5 can be represented as below, a region belonging to -3 and 5 is the region of possible x including 5 and excluding -3.

 Frequently Used Inequalities

1. $(x-a)(x-b)<0 \Rightarrow x \in(a, b)$, where $a<b$
2. $(x-a)(x-b)>0 \Rightarrow x \in(-\infty, a) \cup(b, \infty)$, where $a<b$
3. $x^2 \leq a^2 \Rightarrow x \in[-a, a]$
4. $x^2 \geq a^2 \Rightarrow x \in(-\infty,-a] \cup[a, \infty)$