Problem 12

Question

$9-16$ State whether each inequality is true or false. $$ -\frac{1}{2}<-1 $$

Step-by-Step Solution

Verified

Answer

False

1Step 1: Understand the Comparison

We need to compare two numbers: 1. $-\frac{1}{2}$, which is negative one-half.2. $-1$, which is negative one.Determine if $-\frac{1}{2}$ is less than $-1$.

2Step 2: Analyze the Placement on a Number Line

On a number line, the further to the left a number is, the smaller its value. Both numbers are negative, so analyze which is further left:- $-1$ is further to the left compared to $-\frac{1}{2}$ on a number line.- Thus, $-1$ is less than $-\frac{1}{2}$.

3Step 3: Conclusion

Based on the analysis, we conclude that $-\frac{1}{2}$ is not to the left of $-1$. Therefore, $-\frac{1}{2} < -1$ is false.

Key Concepts

Number lineNegative numbersAlgebraic expressions

Number line

A number line is a visual representation of numbers laid out in a straight line. This tool is extremely helpful in understanding algebraic concepts and inequalities. Each point on the line corresponds to a number, with all numbers arranged from left to right in increasing order.When working with inequalities, you can determine the size of numbers by their positions on the number line. Numbers to the left are smaller, and numbers to the right are larger. For example:

Negative numbers, such as $-1$ and $-\frac{1}{2}$, are found to the left of zero.
Zero is the central point, separating negative numbers from positive numbers.
As you move right, the numbers increase in value.

When comparing numbers or expressions on a number line, their positions help determine inequalities. In the example $-\frac{1}{2} < -1$, we conclude by their positions that $-\frac{1}{2}$ is actually to the right of $-1$, making $-\frac{1}{2} < -1$ false.

Negative numbers

Negative numbers are values less than zero. They appear on the left side of the number line and are a crucial component in algebra and arithmetic. Understanding negative numbers is important when solving inequalities and algebraic expressions.Here are some important points:

Negative numbers are denoted by a minus $(-$) sign in front of a numeral.
The further a negative number is from zero, the smaller its value.
Comparing negative numbers often seems counterintuitive: e.g., $-1$ is less than $-\frac{1}{2}$ because it is further left on the number line.

Negative numbers come up often in algebra, and recognizing their behaviors and properties is essential for solving equations and inequalities alike.

Algebraic expressions

Algebraic expressions contain variables, numbers, and operations. They can represent a constant value or an inequality and are foundational to algebra.Key elements of algebraic expressions include:

Terms: Individual components separated by plus $(+)$ or minus $(-$) signs.
Coefficients: Numbers used to multiply variables.
Variables: Symbols representing numbers, often denoted by $x, y, z$.
Constants: Numbers that stand alone without variables.

Solving inequalities often involves working with algebraic expressions. Consider expressions like $-\frac{1}{2}$ and $-1$, which are also simple inequalities. Understanding their structure helps in comparing them effectively using rules of algebra, such as those regarding negative numbers on a number line.

Problem 12

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Problem 13

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