Question

How do you solve \[4x+3 < 6x+7\] ?

Answer 1

Hint: This type of question is based on inequalities and can be solved like a linear equation by subtracting, adding, multiplying, and dividing terms on both sides. We try to do operations on the equation such that the x terms are on one side and constants on the other, so that we can get a clear range for x. Here, subtract $4x+7$ on both sides and then try to find a range for x by dividing both sides with 2.
The following two points are used in our solution:
(i) Adding and subtracting terms on both sides does not change the inequality.
(ii) Multiplying positive reals on both sides does not change the inequality.

Complete step by step answer:
We have been asked to solve the \[4x+3 < 6x+7\] equation.
We solve this equation by finding a range for x.
We subtract $4x+7$ from both sides of the equation, we get:
\[4x+3-(4x+7) < 6x+7-(4x+7)....(i)\]
$4x+7$ is chosen so that the x terms are on one side and constants on the other.
From (i)
$\Rightarrow -4 < 2x....(ii)$
We divide both sides by 2, which does not change the inequality as 2 is a positive real. So, dividing both sides by 2, we get:
$\Rightarrow -2 < x$
This is also done to remove 2, which is a constant, from the right-hand side of (ii).
Therefore, \[4x+3 < 6x+7\] holds for all x greater than -2.
This can also be written as $x\in \left( -2,\infty \right)$ .

Note: A common mistake one makes often is to think that multiplying any term on both sides does not change the inequality, whereas multiplying negative terms flips the inequality. Another way one can think about this is that taking terms from right to left or left to right changes addition to subtraction and vice versa and taking positive terms from right to left or left to right changes division to multiplication and vice versa.