Question

The set A = \left\\{ {x:|2x + 3| < 7} \right\\} is equal to the set:
A.B = \left\\{ {x: - 3 < x < 7} \right\\}
B.C = \left\\{ {x: - 13 < 2x < 4} \right\\}
C.D = \left\\{ {x:0 < x + 5 < 7} \right\\}
D.E = \left\\{ {x: - 7 < x < 7} \right\\}

Answer 1

Here, we are required to find the set among the given options which is equal to the given set $A$. We will convert the given set into roster form and then solve for the inequality. We will remove the modulus and solve the inequality to its simplest form to get the required set, which when again written in set-builder form, will be equal to one of the sets present in the options.

Complete step-by-step answer:
The given set is A = \left\\{ {x:|2x + 3| < 7} \right\\}.
Now, first of all, we will solve the inequality $|2x + 3| < 7$.
Here, we have the ‘modulus sign’ on the left hand side of the inequality.
Modulus means that we have to take the absolute value of the terms present inside it. We will only take the non-negative values of the terms present inside the modulus when we will remove it.
On the contrary, while removing the modulus sign, we take into consideration both the positive and negative values of the terms present on the other side of the inequality.
Hence, when we will remove the modulus of the inequality $|2x + 3| < 7$.
It can be written as:
$- 7 < 2x + 3 < 7$
Now, subtracting 3 on each side, we get
$\Rightarrow - 7 - 3 < 2x + 3 - 3 < 7 - 3$
$\Rightarrow - 10 < 2x < 4$
Dividing by 2 on each side, we get
$\Rightarrow - 5 < x < 2$
Now, adding 5 on each side, we get
$\Rightarrow - 5 + 5 < x + 5 < 2 + 5$
$\Rightarrow 0 < x + 5 < 7$
Hence, this set can be written as \left\\{ {x:0 < x + 5 < 7} \right\\}
Hence, the set A = \left\\{ {x:|2x + 3| < 7} \right\\} is equal to the set D = \left\\{ {x:0 < x + 5 < 7} \right\\}.
Therefore, option C is the correct answer.

Note: In mathematics, a set consists of a list of elements or numbers which are enclosed in curly brackets. A set can be written in two forms, i.e. set-builder form or the roster form.
Set-builder form is used to represent an equation, an inequality, or the numbers that have some kind of relation. This is also used to represent an infinite number of elements.
The roster form is the simpler form. In this form, we separate the numbers with the help of commas and they are enclosed again, in brackets.
Usually in a question, we are given set-builder form and to solve it further, we convert it to roster form. This makes the question easier to solve.