Problem 89 - Algebra 2

Question

DECORATION. Samantha is going to use more than 75 but less than 100 bricks to make a patio off her back porch. If each brick costs \(\$ 2.75,\) write and solve a compound inequality to determine the amount she will spend on bricks.

Step-by-Step Solution

Answer

Samantha will spend between \(\$206.25\) and \(\$275\) on bricks.

Step 1: Understanding the Range of Bricks

Samantha will use more than 75 but less than 100 bricks. This can be expressed as the compound inequality: \[ 75 < x < 100 \] where \( x \) is the number of bricks.

Step 2: Setting Up the Cost Equation

Each brick costs \(\$2.75\). Therefore, the total cost \( C \) for \( x \) bricks can be represented by the equation: \[ C = 2.75x \] We will use this equation to find the range of total costs.

Step 3: Writing the Compound Inequality for Cost

Using the inequality \( 75 < x < 100 \), substitute \( x \) in the cost equation to form a new inequality: \[ 75 < x < 100 \Rightarrow 75 \cdot 2.75 < C < 100 \cdot 2.75 \]

Step 4: Solving the Inequality

Calculate the values: \[ 75 \cdot 2.75 = 206.25 \] \[ 100 \cdot 2.75 = 275 \] So, the inequality becomes: \[ 206.25 < C < 275 \]

Step 5: Finalizing the Solution

Thus, the amount Samantha will spend on bricks is greater than \(\\(206.25\) and less than \(\\)275\).

Key Concepts

Range of Values

Understanding the range of values involves determining the set of possible quantities or amounts that meet given conditions. In Samantha's case, she aims to use more than 75 but less than 100 bricks for her patio. This means any number of bricks she uses must fall within the open interval from 75 to 100. We express this situation using a compound inequality:

• The lower bound is 75.
• The upper bound is 100.

In inequality form, it's written as \( 75 < x < 100 \), where \( x \) is the number of bricks. "More than 75" translates to \( x > 75 \), and "less than 100" to \( x < 100 \). This range tells us that while 76 and 99 are acceptable values, 75 and 100 themselves are not. These boundaries help us determine the subsequent calculations for cost.

Cost Calculation

Cost calculation involves determining the total expense incurred for a particular quantity of items, based on their individual price. Every brick Samantha buys costs \( \\(2.75 \). To find out the total cost for \( x \) bricks, we use the formula:

• \[ C = 2.75x \]

This equation allows us to calculate the total expenditure by multiplying the price per brick by the number of bricks. Once we have the range of \( x \) from the compound inequality \( 75 < x < 100 \), we apply this range to find the corresponding cost range:

• Substitute 75 and 100 into \( C = 2.75x \).
• This gives us \( 206.25 < C < 275 \).

So, the total cost will fall somewhere between \( \\)206.25 \) and \( \$275.00 \). This range reflects the potential expenditure, considering all possible quantities of bricks that Samantha might purchase.

Inequalities in Mathematics

Inequalities are mathematical expressions that depict the relationship between two values that are not necessarily equal. They use inequality signs, such as:

• "<" (less than)
• "<=" (less than or equal to)
• ">" (greater than)
• ">=" (greater than or equal to)

Compound inequalities, like \( 75 < x < 100 \), show two conditions at once. Here, \( x \) must satisfy both inequalities simultaneously. Understanding inequalities helps us determine boundaries or limits, as in Samantha's case with the number of bricks. When used in cost calculations, inequalities provide a range of possible values rather than pinpointing a single outcome, allowing for dynamic decision-making within specified parameters.