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Chapter 3: Problem 15

Solve each inequality. $$-1>-y$$

Short Answer

Expert verified
1 < y, or y > 1.

Step by step solution

01

- Simplify the inequality

Begin by simplifying the given inequality \(-1 > -y\)
02

- Multiply both sides by -1 to isolate y

To isolate \(y\), multiply both sides of the inequality by 鈭�1. Note: When multiplying an inequality by a negative number, the direction of the inequality symbol reverses. \(-1 > -y \) becomes \( 1 < y\)

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Key Concepts

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

algebraic inequalities
Inequalities are like equations but instead of an equal sign, they use inequality symbols such as <, >, 鈮�, and 鈮�. Solving inequalities involves finding the range of values that make the inequality true.

In our example, the inequality is \(-1 > -y\). We need to find the values of y that satisfy this condition.

To simplify, treat the inequality much like you would an equation. The difference is we will use properties of inequalities instead of just performing operations on both sides.
inverse operations
Inverse operations are mathematical operations that reverse the effect of another operation. Addition and subtraction are inverse operations, just like multiplication and division.

In solving \(-1 > -y\), we need to isolate the variable y. To remove the negative sign in front of y, we multiply both sides by -1.

But remember, multiplying or dividing both sides of an inequality by a negative number reverses the inequality direction.

So, when we multiply \(-1 > -y\) by -1, it changes to \(1 < y\). Now we have y on one side and the inequality points in the correct direction.
negative numbers in inequalities
Handling negative numbers in inequalities is tricky because the direction of the inequality reverses when you multiply or divide by a negative number.

This is a crucial rule that you'll use often in algebra.

For example, in \(-1 > -y\), we multiply both sides by -1 to get rid of the negative sign in front of y. Remember:
  • > becomes <
  • < becomes >
  • >= becomes <=
  • <= becomes >=

This flips \(-1 > -y\) to \(1 < y\), which is the simplified form of the inequality.

Always keep an eye out for negative numbers in inequalities and remember this rule.

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Most popular questions from this chapter

Summarize the properties of equality in one sentence.

Evaluate the given expression. $$\frac{-15-5}{-2(-5)(-2)}$$

Perform the indicated operations on the given inequality. Sketch the resulting inequality on a number line. \(-4 \leq 4 y \leq 12 ; \quad\) divide each member by 4

A Web site manager notices that on a certain day her Web site experienced \(12,468\) hits. She separates the hits into three categories: Category 1: Hits that spend less than 1 minute on the Web site. Category 2: Hits that spend between 1 minute and 5 minutes on the Web site. Category 3: Hits that spend more than 5 minutes on the Web site. There were twice as many hits in category 2 as in category \(3,\) and the total number in category 1 is equal to the numbers in categories 2 and 3 together. How many hits were in each category?

Perform the indicated operations on the given inequality. Sketch the resulting inequality on a number line. \(-x<-3 ;\) multiply each side by \(-1\)
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