91Ó°ÊÓ

When facing a complex equation, the first step is to simplify it to its most straightforward form. Simplifying an equation means to reduce its complexity, making it easier to solve. This process involves combining like terms, which are terms that have the same variable raised to the same power. For our example,

\[ \frac{-(4 x+3-5 x)}{3}=2 \]
we combine the terms with 'x' to get a simplified equation. The minus sign outside the brackets affects every term inside, leading to

\[ \frac{-(-x + 3)}{3} = 2. \]
Here, the like terms '4x' and '-5x' combine to '-x', and the operation between numbers inside the brackets is simplified to '+3'. Notice that 'simplifying' does not mean 'solving', but rather preparing the equation to be easier to solve in the subsequent steps.

Isolating the variable is essential in solving an equation. It involves getting the variable on one side of the equation and all other terms on the opposite side. This provides a clear solution for the variable. To isolate the variable in our example,

\[ -x + 3 = 6, \]
we need to separate 'x' from the number '3'. We do this by performing the same operation on both sides of the equation, maintaining the equality. Subtracting '3' from both sides yields

\[ -x = 3, \]
However, we have '-x' instead of 'x', so we multiply by '-1' to obtain

\[ x = -3. \]
By doing so, we isolate 'x' and solve for its value. It's vital to remember that whatever we do to one side, we must do to the other to preserve the equation's balance.

Solving inequalities is similar to solving equations but with a focus on determining the range of values for which the inequality holds true. Unlike equations which have a singular solution, inequalities can have a set of solutions. The main steps in solving inequalities include simplifying the inequality, isolating the variable, and then considering the direction of the inequality.

When multiplying or dividing both sides of an inequality by a negative number, it is crucial to remember to reverse the direction of the inequality. For example, if we had an inequality like

\[ -x > 3, \]
the solution would be obtained by dividing both sides by '-1', which in turn would reverse the inequality sign, giving us

\[ x < -3. \]
It's essential to pay attention to this detail to arrive at the correct set of solutions for an inequality.