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Chapter 5: Problem 28

For the following problems, solve the inequalities. $$ -3 y>39 $$

Short Answer

Expert verified
Answer: For the given inequality, the range of values for y is y < -13.

Step by step solution

01

Divide by (-3)

To isolate the variable y, divide both sides of the inequality by (-3): $$ \frac{-3y}{-3}>\frac{39}{-3} $$
02

Simplify and Flip the Inequality Symbol

Since we divided by a negative number, flip the inequality symbol, and simplify the fractions: $$ y<\frac{-39}{3} $$ And $$ y<-13 $$ So the solution to the given inequality is: $$ y<-13 $$

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Key Concepts

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

Solving Inequalities
Solving inequalities is much like solving equations, with just a few extra rules to keep in mind. Inequalities are mathematical expressions where two values are compared using inequality symbols such as less than (<), greater than (>), less than or equal to (鈮�), and greater than or equal to (鈮�).

To solve an inequality, we follow similar steps to solving an equation:
  • First, isolate the variable you are solving for on one side of the inequality.
  • Next, perform the same operations on both sides of the inequality to keep it balanced.
  • Finally, simplify the expression to find the solution set, often a range of values.
In our example, \(-3y > 39\), we isolate \(y\) by dividing both sides by \(-3\), which brings us to the next important concept about handling negative numbers.
Negative Numbers
Negative numbers are numbers less than zero, and they can be a little tricky when solving inequalities. They are denoted by a minus sign (-). When operating with negative numbers, there are specific rules to keep in mind:

  • Adding a negative number is the same as subtracting its absolute value.
  • Subtracting a negative number is the same as adding its absolute value.
  • Multiplying or dividing two negative numbers gives a positive result.
  • Multiplying or dividing a positive and a negative number gives a negative result.
In our inequality \(-3y > 39\), we divide by \(-3\) to make \(y\) positive. This brings us to our next rule when dividing by a negative number.
Flipping Inequality Sign
A unique rule to remember when working with inequalities involves the direction of the inequality sign. When an inequality is multiplied or divided by a negative number, the inequality sign must be flipped. This is because the relative ordering of the numbers on the real number line reverses under multiplication or division by negative numbers.

Let's see how this works with our example:
  • Initially, we have: \(-3y > 39\).
  • To isolate \(y\), we divide both sides by \(-3\), resulting in:\(y < -13\).
  • Notice how the "greater than" symbol (>) changed to "less than" (<) after dividing by the negative number.
By understanding this rule, you can accurately solve inequalities and interpret their solutions, ensuring you know how the sign's direction plays a vital role in defining the solution set.

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