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Chapter 14: Problem 58

Solve and write the answer in interval notation. $$-y<4$$

Short Answer

Expert verified
The solution in interval notation is (-4, +鈭�).

Step by step solution

01

Solve the inequality

The inequality is currently written as -y < 4. We want to solve for y. To do this, we divide all parts of the inequality by -1. However, when dividing or multiplying an inequality by a negative number, the direction of the inequality sign must be flipped. So we get y > -4.
02

Convert solution to interval notation

Interval notation is a way of representing continuous sets of real numbers. Since y > -4, this means that y can be any number greater than -4. In interval notation, this is written as (-4, +鈭�).

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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 efficient way to describe a range of values, typically numbers. It uses parentheses \((\) and brackets \([\) to show inclusivity or exclusivity of endpoints.

For example:
  • Parentheses \((\) mean the endpoint is not included. So (-4, +鈭�) means all numbers greater than -4, but not -4 itself.
  • Brackets \([\) mean the endpoint is included. So [-4, +鈭�) would mean -4 is included.
This notation is often used to represent solutions to inequalities in mathematics, providing a clear visual of which numbers are included in the solution set.
Inequalities Solving
Solving inequalities involves finding all values of the variable that make the inequality true. The overall process is similar to solving equations, but with an important twist: the inequality direction flips when you multiply or divide by a negative number.

Here's how to approach solving an inequality:
  • Isolate the variable on one side. For -y < 4, divide by -1 to solve for y.
  • Remember, when dividing by a negative number, flip the inequality sign. So -y < 4 becomes y > -4.
  • Check the solution by plugging values back into the original inequality.
These steps help ensure the inequality is correctly solved and can be confidently expressed in interval notation.
Negative Number Multiplication
Multiplying or dividing by a negative number when solving inequalities requires special attention because it reverses the inequality sign.

Key points to keep in mind:
  • If you multiply or divide both sides of an inequality by a negative number, flip the inequality sign.
  • For example, with -y < 4, dividing by -1 flips it to y > -4.
  • Remember this principle anytime a negative number is involved in an inequality.
This rule ensures the inequality continues to represent the correct range of solutions.
Real Numbers Representation
Real numbers include all the numbers along the number line, covering everything from fractions to irrational numbers.

When representing real numbers in the context of inequalities:
  • Use intervals to show continuous sets. For y > -4, the interval is (-4, +鈭�), capturing all real numbers greater than -4.
  • An understanding of intervals helps in visualizing solutions to inequalities.
  • Intervals tell us precisely which numbers satisfy the inequality conditions.
Using real numbers and interval notation together allows for a clearer, more precise mathematical communication.

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