Problem 43 - Algebra A Combined Function

Question

Graph each set of numbers on the number line. $$ -4.5, \frac{7}{4}, 3.25,-\frac{3}{2} $$

Step-by-Step Solution

Answer

Mark -4.5, 1.75, 3.25, and -1.5 on the number line.

Step 1: Draw a Number Line

Begin by drawing a horizontal line, which will represent the number line. Mark the center of the line as 0, and make sure to include negative numbers to the left and positive numbers to the right. Choose an appropriate scale, such as marking every whole number.

Step 2: Locate -4.5 on the Number Line

Move to the left of 0 and count 4.5 units to find -4.5. Mark the point clearly on the number line, 4.5 units left from the origin 0.

Step 3: Convert \( \frac{7}{4} \) to Decimal

Convert the fraction \( \frac{7}{4} \) to a decimal for easier plotting. Divide 7 by 4 to get 1.75. This will help you place the number on the number line.

Step 4: Locate \( \frac{7}{4} \) (or 1.75) on the Number Line

Start from 0 and move 1.75 units to the right. Mark this position on your number line, as it corresponds to \( \frac{7}{4} \).

Step 5: Locate 3.25 on the Number Line

From 0, count and move 3.25 units to the right. This is straightforward as it's already a decimal. Mark 3.25 clearly on the number line.

Step 6: Convert \( -\frac{3}{2} \) to Decimal

Convert the fraction \( -\frac{3}{2} \) to a decimal form. Divide 3 by 2 to get 1.5, so \( -\frac{3}{2} \) is -1.5.

Step 7: Locate \( -\frac{3}{2} \) (or -1.5) on the Number Line

From 0, count and move 1.5 units to the left. Mark this position clearly to represent \( -\frac{3}{2} \).

Key Concepts

Number Line

A number line is a visual representation of numbers laid out in a sequential line where each point corresponds to a number. This line helps us understand numbers better by showing their order and spacing.
On a traditional number line:

• The center of a number line is always zero (0).
• Numbers to the right of zero are positive, displayed in ascending order.
• Numbers to the left of zero are negative, also in ascending order but with a minus sign in front, like -1, -2, etc.
• The number line can extend infinitely in both directions.

Marking numbers on the number line allows for easy comparison and helps us see relationships among numbers. It's a fundamental tool in math for operations like addition and subtraction, and for demonstrating concepts like greater than or less than.

Fraction to Decimal Conversion

Fractions and decimals are two different ways to represent numbers, and sometimes it's useful to convert between them for ease of understanding or comparison.
The calculation is straightforward:

• To convert a fraction like \( \frac{7}{4} \) to a decimal, simply divide the numerator (7) by the denominator (4). This gives us 1.75.
• Similarly, when converting \( -\frac{3}{2} \), divide 3 by 2 to get 1.5 and apply the negative sign, resulting in -1.5.

Decimals can facilitate plotting numbers on a number line. They show more precise locations, especially when dealing with values between whole numbers.

Positive and Negative Numbers

Positive and negative numbers are the building blocks on a number line, each representing direction and distance from zero.

• **Positive numbers** are all the numbers greater than zero. They are located to the right of zero on a number line.
• **Negative numbers** are less than zero and appear to the left of zero. They have a minus (-) sign in front.
• Zero is neutral, neither positive nor negative, and acts as the central dividing point on the number line.

Understanding these allows us to perform operations such as addition and subtraction correctly, as they help determine the direction we move on the number line.

Locating Points on the Number Line

Locating numbers on a number line is an essential skill that requires understanding both the number itself and its place relative to others. Here's a simple method for locating points, using the examples given:

• For \(-4.5\), start at zero and move 4.5 units left, as it is negative. This places the point accurately on the line.
• For 1.75, again start at zero, but this time move 1.75 units right, since it is a positive number.
• With 3.25, also start at zero and proceed 3.25 units right.
• Finally, \(-1.5\) involves starting at zero and moving 1.5 units left due to the negative sign.

Marking these points along the number line helps visualize the values and understand relationships between different numbers. This can be indispensably useful in problem-solving and mathematical reasoning.