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Chapter 14: Problem 53

Solve and write the answer in interval notation. $$-5 y \geq 0$$

Short Answer

Expert verified
The solution in interval notation is \((-\infty, 0]\)

Step by step solution

01

Isolate the variable

To isolate the variable \(y\), divide both sides of the equation by -5. Remember when dividing by a negative number, the sign of the inequality changes to its opposite.
02

Perform the division

So, the inequality becomes \(y \leq 0\).
03

Write the answer in interval notation

Solution in interval notation would be \((-\infty, 0]\). That means \(y\) belongs to all numbers less than or equal to 0.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Interval Notation
Interval notation is a concise and structured way to describe a set of numbers. It uses parentheses or brackets to convey which numbers are included in or excluded from a set. Here's how it works:

  • Parentheses ( ) are used for numbers that are not included in the set. This is also called an open interval.
  • Brackets [ ] are used for numbers that are included in the set, known as a closed interval.
In the solution to this inequality, we use the interval notation \((-\infty, 0]\). The \(\infty\) sign means infinity, indicating that the set of numbers extends indefinitely. Since infinity is not a specific number, we always use a parenthesis with it. The 0 has a bracket indicating that 0 is included in the set. This notation efficiently illustrates that y includes all numbers less than or equal to zero.
Negative Coefficient
A negative coefficient in an inequality affects how you solve it. Coefficients are the numbers that multiply the variable. In the given exercise, \(-5\) is a negative coefficient multiplying the variable \(y\).

  • When you divide or multiply both sides of an inequality by a negative number, you must reverse the inequality sign. This is crucial because it ensures the inequality still reflects the correct order of the numbers.
In our problem, dividing by \(-5\) flips the inequality from \(\geq\) to \(\leq\). This change maintains the correct relationship between \(y\) and the numbers it includes in its solution set. Remember, always flip the sign when dividing by a negative number to avoid mistakes!
Variable Isolation
Isolating the variable means transforming an equation or inequality so that the variable is by itself on one side. This helps to identify the values the variable can represent.

  • Start by performing arithmetic operations to get the variable by itself. For inequalities involving negative coefficients, you might divide both sides by the negative number.
For the given example, isolating the variable \(y\) involved dividing both sides of the inequality \(-5y \geq 0\) by \(-5\). The key step here was recognizing that division by a negative flips the inequality sign. Doing so simplified the inequality to \(y \leq 0\), which sets up nicely to express the solution using interval notation.

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