Question

Graph the numbers on a number line. Then write two inequalities that compare the two numbers. $$-0.5 \text { and }-\frac{1}{3}$$

Short answer

-0.5 < -1/3 and -1/3 > -0.5

Step-by-step solution

Understand the Numbers

We are given two numbers -0.5 and -1/3. It can be understood better if we convert -1/3 into decimal which gives us approximately -0.33.

Plot on Number Line

Plot the numbers -0.5 and -0.33 on a number line. It is clear that -0.33 is to the right of -0.5, hence it is greater.

Write Inequalities

Since -0.33 is greater than -0.5, we can write the inequalities as -0.5 < -1/3 and -1/3 > -0.5.

Key concepts

Plotting Rational Numbers

When it comes to plotting rational numbers on a number line, clarity is key. Rational numbers, which include fractions and decimals, represent a value that can be expressed as the quotient of two integers. To correctly plot these numbers, first, ensure that the number line is accurately labeled with appropriate intervals.

For example, when comparing the numbers -0.5 and \( -\frac{1}{3} \), we must first find a common scale. Both -0.5 and \( -\frac{1}{3} \) are less than zero, so they will be placed to the left of the number line's origin. The key is to understand their position relative to each other, as well as to other rational numbers that might already be plotted on the line.

To ease our understanding, it is helpful to think of rational numbers in terms of money or measurement, which are typically divided into fractional or decimal components. Just like you can measure distances or count change, rational numbers can be located on a number line, dividing them into smaller, manageable parts until we find their exact position.

Converting Fractions to Decimals

Before plotting points on a number line, converting fractions to decimals can streamline the comparison process. This is because decimals are often easier to interpret at a glance, especially when dealing with negative values.

For conversion, divide the numerator (the top number) of the fraction by its denominator (the bottom number). With \( -\frac{1}{3} \), you would divide 1 by 3, yielding approximately -0.33. Converting fractions to decimals thus simplifies the process of identifying which of two rational numbers is greater or smaller when plotted on a number line.

Helpful Conversion Tips:

• Always retain the sign of the original fraction when converting.
• For recurring decimals, use an approximation or, if precision is required, a bar over the recurring digit(s).
• Use a calculator if the division seems too complex or to verify your result.

Writing Inequalities

Conveying the relationship between numbers succinctly is the purpose of writing inequalities. After plotting rational numbers, we can visually compare their positions and translate this observation into an inequality.

For the numbers -0.5 and -0.33, we can state that -0.5 is less than -0.33, or -0.5 < -0.33. Conversely, -0.33 is greater than -0.5, expressed as -0.33 > -0.5. Writing inequalities involves using the symbols '<' and '>', which denote 'less than' and 'greater than,' respectively. To avoid confusion and correctly portray the relationship, remember that the small end of the inequality symbol always points to the smaller number while the wide end points to the larger number.

Key Takeaways for Writing Inequalities:

• The inequality sign should always point to the smaller number.
• When dealing with negative numbers, a number closer to zero is considered larger.
• Inequalities can also be written using '≤' or '≥' if the values are equal or include the endpoints of a range.