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Chapter 14: Problem 54

Solve and write the answer in interval notation. $$-3 z<0$$

Short Answer

Expert verified
The solution to the inequality -3z < 0 in interval notation is (0, +鈭�).

Step by step solution

01

Isolate 'z' in inequality

To isolate 'z', divide both sides by -3. The inequality switches direction whenever multiplying or dividing by a negative number. So, the inequality -3z < 0 becomes z > 0 when divided by -3.
02

Write the solution in interval notation

Interval notation is a way of expressing the solution set for an inequality. The solution z > 0 covers all real numbers greater than 0. Therefore, in interval notation, the solution is expressed as (0, +鈭�). The '(' indicates that 0 isn't included in the interval, and '+鈭�' suggests that it stretches indefinitely to positive infinity.

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Key Concepts

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

Interval Notation
Understanding interval notation is key to expressing the solutions to inequalities concisely. When solving inequalities, we often describe a range of numbers that satisfy the condition, rather than a single number. Interval notation helps us to communicate this range.
  • Use parentheses, \((\), to indicate that an endpoint is not included in the solution.
  • Use brackets, \([\), to indicate that an endpoint is included in the solution.
  • An open interval, like \((0, +\infty)\), expresses numbers greater than 0 but not equal to 0.
  • A closed interval, like \([0, 1]\), includes both 0 and 1.
Interval notation is efficient for representing solutions to inequalities because it is shortened and eliminates the need for lengthy verbal descriptions. It's important to remember the infinity symbol, \(\infty\), is always used with a parenthesis because infinity is not a number that can be reached or enclosed.
Solving Inequalities
Solving inequalities resembles solving equations but with extra rules when dealing with negative numbers. In the given example, we have the inequality \[-3z < 0\]The goal is to isolate the variable, \(z\). We divide each side of the inequality by -3. However, it鈥檚 crucial to note that whenever we multiply or divide both sides of an inequality by a negative number, the inequality sign flips. This results in \[z > 0\]Here are some key points to remember when solving inequalities:
  • Perform the same operation on both sides of the inequality to maintain balance, much like with equations.
  • Be cautious about flipping the inequality sign when multiplying or dividing by a negative number.
This rule of reversal ensures the integrity of the inequality, providing an accurate solution set.
Negative Coefficient
Dealing with negative coefficients in inequalities brings a unique twist compared to standard equations. A coefficient is the number multiplied by a variable, and in our case, -3 is the negative coefficient for \(z\).
When we perform operations on inequalities with negative coefficients, we must pay careful attention:
  • Dividing or multiplying the entire inequality by a negative coefficient requires the inequality sign to be reversed. This is because the relative order of the numbers changes when multiplied by a negative.
  • For example, if -3\(z\) is less than 0, and we resolve this by dividing by -3, the inequality becomes \(z > 0\).
  • When solving, always check if a variable term has a negative coefficient and proceed with caution.
These principles ensure that the final solution correctly represents the set of numbers satisfying the inequality. Understanding the role of negative coefficients is fundamental to mastering inequalities.

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