Question

Solve the inequality \[2x-5\le \dfrac{\left( 4x-7 \right)}{3}\] :
1. \[x\in \left( -\infty ,4 \right)\]
2. \[x\in \left( -\infty ,4 \right]\]
3. \[x\in \left( -\infty ,8 \right]\]
4. \[x\in \left( -\infty ,-4 \right]\]

Answer 1

Hint: This type of question is based on linear inequalities, and we try to execute operations on the equation such that terms are on one side and constants are on the other, so that we can have a clear range of \[x\], and then you check which choice in the above supplied possibilities is accurate.


Complete step by step answer:
In mathematics, inequality occurs when two mathematical statements or two numbers are compared in a non-equal way. Inequalities are classified as either numerical or algebraic inequalities. When two numbers are compared on a number line based on their value, numerical inequality occurs. When one expression is more than or less than another, it is called algebraic inequality. Different types of disparities can be represented in a variety of ways. Let's learn about linear inequalities, solving linear inequalities, graphing linear inequalities, and linear inequalities with one variable, as well as graphing one variable, in this post. Linear inequalities are expressions in which the inequality symbols \[<,>,\le ,\ge \] are used to compare any two values. These numbers could be numerical, algebraic, or a mix of the two.
Laws of Inequality:
The sign of inequality does not change if the same number can be added or removed from both sides of the inequality.
 Adding or subtracting a positive number from each side of an inequality has no effect on the inequality.
 Reverse the inequality by multiplying or dividing either side by a negative value.

To solve an inequality, we can:
Add (or subtract) the same amount to (from) both sides while keeping the inequality's sign the same.
Without changing the sign of the inequality, multiply (or divide) both sides by the same positive number. The sign of inequality is reversed if both sides of the inequality are multiplied (or divided) by the same negative quantity that is \[>\] becomes \[<\] and vice versa.
Now according to the question:
We have given the inequality
\[\Rightarrow 2x-5\le \dfrac{\left( 4x-7 \right)}{3}\]
We have to solve this equation by finding a range for \[x\]
\[\Rightarrow 3\times \left( 2x-5 \right)\le \left( 4x-7 \right)\]
\[\Rightarrow 6x-15\le 4x-7\]
\[\Rightarrow 6x-4x\le 15-7\]
\[\Rightarrow 2x\le 8\]
\[\Rightarrow x\le \dfrac{8}{2}\]
\[\Rightarrow x\le 4\]
Hence \[x\] lies between \[x\in \left( -\infty ,4 \right]\]

So, the correct answer is “Option 2”.

Note:
Students used to make a common mistake while solving a linear inequality and a linear equation; the only difference in solving an equation and an inequality is that while multiplying or dividing by a negative number the sign of inequality should be reversed but this does not happen in case of equation.