91Ó°ÊÓ

Solving inequalities is similar to solving equations, but with one crucial difference: the inequality sign. In the given exercise, the goal is to isolate the variable, which is "y" in this case, to determine the range of values it can take.
The inequality given is a linear inequality because it involves a first-degree term in the variable. To solve it:

• You perform operations just like you would with an equation - adding, subtracting, multiplying, or dividing both sides.
• It's important to simplify the inequality step-by-step, maintaining balance on both sides.
• As with equations, whatever operation you apply to one side, you must apply it to the other to keep the statement true.

By systematically isolating "y", we get the inequality in its simplest form, showing all the possible values "y" could take. With the final answer, the values of "y" lie within a designated range.

Understanding algebraic expressions is crucial when dealing with inequalities. An algebraic expression combines numbers, variables, and arithmetic operations. For example, in the inequality \(-11y + 4 > 15\), "-11y + 4" is an algebraic expression.
The key is to interpret these expressions:

• The term "-11y" includes the coefficient (-11) attached to the variable (y); this indicates the rate of change.
• The constant "4" indicates a fixed value added to the expression.
• When manipulating the expressions, aim to isolate the variable and restructure the expression into a more solvable form.

The manipulation of these sums or differences is a foundation for solving more complex equations and inequalities. Managing the parts of an expression allows you to correctly and efficiently solve the inequality.

One unique feature of solving inequalities is the rule concerning the inequality sign flipping. When solving inequalities, like the one in the exercise, this rule is particularly important.
Whenever you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. This happens because the relationship between the sides of the inequality changes direction when multiplied or divided by negatives.

• For instance, in the step where we divided by -11, the inequality \(-11y > 11\) became \(y < -1\) because of the flipping rule.
• This rule helps keep the inequality true, ensuring we correctly represent the range of values the variable can take.

Always remember, flipping is only needed when dealing with negative multipliers or divisors. It's a common source of mistakes, so watch out for negative operations. This rule ensures the solution to the inequality accurately reflects all possible values that satisfy it.