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Inequalities - Practice Questions & MCQ
Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main
Quick Facts
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32 Questions around this concept.
Solve by difficulty
Which values of $x$ satisfy the inequality $(x+1)(x-3)<0$?
If $x<5$, then
Among the inequalities below, which ones are true for all natural numbers $n$ greater than $1000 ?$
$
\begin{aligned}
& \text { (I) } n!\leq n^n \\
& \text { (II) }(n!)^2 \leq n^n \\
& \text { (III) } 10^2 \leq n! \\
& \text { (IV) } n^2 \leq(2 n)!
\end{aligned}
$
Given that x, y and b are real numbers and $x< y,b< 0$ then
Values of x corresponding to $y=2$ are $x_1$ and $x_2\left(x_1>x_2\right)$ and corresponding to $y=0$ is $x_3$, then $x_1+x_3-x_2$ equals
If $a<b$, then which of the following is always true for $a, b \in R$
The solution of the inequality $2(x-1)>x+4$ is
Number of Integral values of x satisfying the inequality $\frac{x^{2}+5 x-24}{\left |x+4 \right || x-2 |}<0$ is
Concepts Covered - 1
Inequalities
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. $x^2 \geq a^2 \Rightarrow x \in(-\infty,-a] \cup[a, \infty)$
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