Question

SOLVING INEQUALITIES Solve the inequality. $$\frac{x}{6} \leq-2$$

Short answer

The solution to the inequality \( \frac{x}{6} \leq -2 \) is \(x \leq -12\).

Step-by-step solution

Multiplication

First, multiply both sides of the inequality by 6 to isolate x. This gives us \(x \leq -12\).

Solution

So, the solution to the inequality \( \frac{x}{6} \leq -2 \) is \(x \leq -12\). This means that every number less than or equal to -12 is a solution to the problem.

Key concepts

Inequality

When we encounter an inequality, we're dealing with a mathematical expression that shows the relationship between two values that are not necessarily equal. Inequalities can be visualized as a balance scale, where each side might have a different weight. To solve an inequality means to find all the possible values that can make the inequality true. In the given exercise, the task at hand was to solve \( \frac{x}{6} \leq -2 \). The symbol \( \leq \) represents 'less than or equal to'. It's crucial to understand that when we solve an inequality, we are looking for a range of values, not just a single number. This range forms what is known as the solution set, and in the case of the given inequality, any number that is less than or equal to -12 falls within this set.

When it comes to graphing these solutions on a number line, they are often depicted with a solid dot at the border number (if it is included in the range, as indicated by \( \leq \) or \( \geq \) symbols) and a shaded area extending in the direction of the valid values.

Isolate Variable

To isolate the variable means to manipulate the equation or inequality until the variable we are solving for is by itself on one side of the equation or inequality. This is a critical step in solving both equations and inequalities as it helps to clarify what the solution or set of solutions can be for the variable. In our problem \( \frac{x}{6} \leq -2 \), isolating \(x\) involved multiplying both sides by 6. This reverses the division by 6 that \(x\) was subjected to and effectively 'frees' the \(x\) to stand alone. The process of isolating the variable is like peeling layers off an object until you reach the core—the clear, uncomplicated truth of the variable's range of solutions.

It is essential to perform the same operation to both sides of the inequality to maintain its balance. Inequalities demand careful attention because certain operations, like multiplying or dividing by a negative number, will flip the inequality sign.

Multiplication Property of Inequality

The multiplication property of inequality allows us to multiply (or divide) both sides of an inequality by the same positive number without changing the inequality's direction. This is exactly what we did in the step-by-step solution to the exercise when we multiplied both sides by 6. The equivalent algebraic statement went from \( \frac{x}{6} \leq -2 \) to \( x \leq -12 \) without altering the relationship between \(x\) and -12. It's crucial to remember that if we multiply or divide by a negative number, we must flip the inequality sign to maintain the truth of the expression. As it stands, with positive multiplication, the order of the values is preserved.

This property is central in solving inequalities because it allows us to manipulate and simplify the inequality to a more understandable form, without losing any information about the relationship between the values involved. Therefore, a thorough understanding of this property is invaluable for correctly solving and graphing inequalities.