Problem 59 - Algebra 2

Question

To get a chance to win a car, you must guess the number of keys in a jar to within 5 of the actual number. Those who are within this range are given a key to try in the ignition of the car. Suppose there are 587 keys in the jar. Write and solve an equation to determine the highest and lowest guesses that will give contestants a chance to win the car. (Lesson \(1-4 )\)

Step-by-Step Solution

Answer

The lowest guess is 582 and the highest guess is 592.

Step 1: Understand the Problem

We need to determine the range of guesses that are within 5 of the actual number of keys, which is 587. This means the guesses should be in the range from 5 less than 587 to 5 more than 587.

Step 2: Setting Up the Equation for the Range

To find the lowest guess, subtract 5 from the actual number of keys:\[ 587 - 5 \]Similarly, for the highest guess, add 5 to the actual number of keys:\[ 587 + 5 \]

Step 3: Calculate the Lowest Guess

Subtract 5 from 587 to find the lowest number in the range:\[ 587 - 5 = 582 \]

Step 4: Calculate the Highest Guess

Add 5 to 587 to find the highest number in the range:\[ 587 + 5 = 592 \]

Step 5: Determine the Range of Valid Guesses

The valid range of guesses that gives a contestant a chance to win the car is from 582 to 592 inclusive.

Key Concepts

Equation

In algebra, an equation is like a math sentence that shows the relationship between different values. In the context of this exercise, we're tasked with creating an equation that helps us identify a range of potential guesses for the number of keys inside a jar.

To achieve this, let’s think about what the equation should represent. We know that the number of keys has a fixed value of 587, and we're interested in those guesses that fall within 5 keys of this actual amount, whether more or less. This means our equation will revolve around this central number.

The approach to forming this equation is simple: consider the actual number 587 and account for the allowable error, which is 5 keys in either direction. Therefore, two equations can be written:

• Lowest guess equation: \( x = 587 - 5 \)
• Highest guess equation: \( x = 587 + 5 \)

By solving these equations, we identify the valid range for guesses.

Number Line

A number line is a helpful visual tool in understanding inequalities and range. Imagine a straight line marked with numbers, where each point on the line corresponds to a number.

In the context of guessing the number of keys, positioning the numbers 587, 582, and 592 on this line can help in visualizing the acceptable guesses. The number 587 sits at the center, while 582 and 592 mark the boundaries of this acceptable range.

This aids in understanding visually how far we can drift from the actual count. Everything between 582 and 592, inclusive, will fall within the range, showing visually on the number line that any guess falling outside these numbers will not satisfy the condition. It’s as if we draw a line from 582 to 592 on this number graph, indicating all potential guesses for a win.

Algebraic Expressions

Algebraic expressions allow us to generalize specific conditions, such as this guessing game, into simple but powerful statements. These are pieces of math that might include numbers, variables (like \( x \)), and operations like addition or subtraction.

In the exercise at hand, we can form an expression to generalize the rule for winning a key. Suppose \( x \) represents any valid guess. The expression can be written to show valid outcomes:
\( 582 \leq x \leq 592 \)
This expression claims that any value of \( x \) within this range is an acceptable guess.

Utilizing algebraic expressions is vital because it helps to compactly state complex ideas, making solving real-world problems simpler. It also paves the way for easily applying these concepts to other similar guessing challenges, simply by adjusting the numbers in the expression.