Solve 2x−1<4 and −5x−3>−3 and write the solution in interval notation. If there is no solution, type ∅.

Algebraic expressions can be related to each other in many ways. When two expressions are equal to each other, they are called equations and are represented with an equal sign between them (=). When two expressions are not equal, they are called inequations and are represented by non-equal signs between them (>, <, ≥, ≤).

How to solve the given question?

In the question, we are asked to solve the given inequations:

2x - 1 < 4 ... (i), and -5x -3 > -3 ... (ii).

First, we will solve (i), in the following ways:

2x - 1 < 4.

or, 2x - 1 + 1 < 4 + 1 (Adding 1 to both sides of the inequation)

or, 2x < 5 (Simplifying)

or, 2x/2 < 5/2 (Dividing both sides of the inequation by 2)

or, x < 5/2 (Simplifying)

or, x ∈ [tex]\left ( -\infty , \frac{5}{2} \right )[/tex] ...(iii)

Now, we will solve (ii), in the following ways:

-5x -3 > -3.

or, -5x - 3 + 3 > -3 + 3 (Adding 3 to both sides of the inequation)

or, -5x > 0 (Simplifying)

or, -5x/(-5) < 0/(-5) (Dividing both sides of the inequation by -5, sign of the inequality reverses as we are dividing by a negative number)

or, x < 0 (Simplifying)

or, x ∈ [tex]\left ( -\infty , 0 \right )[/tex] ...(iv)

For the solution set, we need the intersection of (iii) and (iv)

[tex]\left ( -\infty , \frac{5}{2} \right )[/tex] ∩ [tex]\left ( -\infty , 0 \right )[/tex]

= [tex]\left ( -\infty , 0 \right )[/tex]

∴ The solution set of the given set of linear inequations 2x - 1 < 4 and -5x - 3 < -3 is [tex]\left ( -\infty , 0 \right )[/tex].

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