Question

If $x$ is real then $3{x^2} + 14x + 11 > 0$, when

A) $x < \- \dfrac{3}{2}$

B) $x < \dfrac{{10}}{3}$ or $x > 1$

C) $x < \- 2$

D) None of the above

Answer 1

Here we have to find the range of $x$. We will factorize the function given in the left side of inequalities. Then we will solve the factors obtained separately, where both the factors will be greater than zero. After solving these two inequalities, we will get the range of $x$ which will satisfy the given inequality. Here we will use the basic properties of inequalities to solve this inequality question.

Complete step-by-step answer:

The given inequality is $3{x^2} + 14x + 11 > 0$.

Write 14x as (11x+3x)

$\Rightarrow$ $3{x^2} + 11x + 3x + 11 > 0$

Simplifying this inequality, we get

$\Rightarrow$ $x\left( {3x + 11} \right) + \left( {3x + 11} \right) > 0$

Taking $\left( {3x + 11} \right)$ common, we get

$\Rightarrow$ $\left( {3x + 11} \right)\left( {x + 1} \right) > 0$

This inequality is possible only when $\left( {3x + 11} \right) > 0$ or $\left( {x + 1} \right) > 0$ and $\left( {3x + 11} \right) < 0$ or $\left( {x + 1} \right) < 0$.

Therefore, the possible values are-

$\Rightarrow$ $x < -\dfrac{{11}}{3}$ or $x > -1$

Thus, the correct option is (D).

Note: Here we have solved the inequality. An inequality represents the relative size of two values.

Some important properties of inequalities are:-

1. Reversal property: If there are two numbers say a and b; and if $a > b$ then we can also write this inequality as $b < a$; where $a > b$ means a is greater than b and $b < a$ means b is less than a.

2. Addition and subtraction property: If there are two numbers say a and b; where $a > b$ and if we add a number say c on both sides of inequality, then the inequality remains the same. i.e. $a + c > b + c$. Similarly, and if we subtract a number say c on both sides of inequality, then the inequality remains the same. i.e. $a - c > b - c$.

3. Multiplication and subtraction property: If there are two numbers say a and b; where $a > b$ and if we multiply a positive number say c on both sides of inequality, then the inequality remains the same i.e. $ac > bc$ but we multiply a negative number then the inequality will not remain the same i.e. $ac < bc$. Similarly, if we divide a positive number say c on both sides of inequality, then the inequality remains the same i.e. $\dfrac{a}{c} > \dfrac{b}{c}$ but we divide a negative number then the inequality will not remain the same i.e. $\dfrac{a}{c} < \dfrac{b}{c}$.