Answer:
When solving an inequality, we must be careful when multiplying or dividing by a variable because doing so can change the direction of the inequality.
For example, consider the inequality:
2x < 6
If we divide both sides by 2, we get:
x < 3
This is the correct solution. However, if we had instead divided both sides by x, we would get:
2 < 6/x
which is not equivalent to the original inequality. To understand why, we can consider two cases:
If x is positive, then dividing both sides by x will preserve the direction of the inequality. However, if x is negative, dividing by x will reverse the direction of the inequality.
If x is zero, then we cannot divide by x, because division by zero is undefined.
In general, when solving inequalities involving variables, it is safer to use algebraic manipulation rules that do not involve multiplying or dividing by variables. These rules include adding or subtracting the same value from both sides, multiplying or dividing both sides by a constant (as long as the constant is positive), and using properties of inequalities such as the transitive property and the addition and multiplication properties.
Hope this helps, I'm sorry if it didn't. If you need more help, ask me! :]