Question

In 1958, the painting Still Life with Apples by Paul Cézanne sold at auction for \(\$ 252,000 .\) The painting was auctioned again in 1993 and sold for \(\$ 28.6\) million. Write a compound inequality that represents the different values that the painting probably was worth between 1958 and 1993

Short answer

The compound inequality that represents the different values that the painting was probably worth between 1958 and 1993 is $252000 \leq x \leq $28600000.

Step-by-step solution

Understand the information given

Firstly, understand that the painting was sold for $252,000 in 1958 and then again for $28,600,000 in 1993. Therefore, it is safe to assume that the value of the painting at any point between these years would have been somewhere between these two amounts.

Write the compound inequality

We can consider these two values as the lower and upper bounds of the painting's worth. Therefore, the inequality to represent all possible values that the painting could have been worth between 1958 and 1993 would be $252000 \leq x \leq $28600000 where x is the value of the painting at a given point in time during this period.

Check your solution

Make sure the inequality correctly shows the range of potential values: x is worth at minimum $252,000 and at maximum $28,600,000. The symbol \( \leq \) denotes that x can also be equal to either of these two values, since we do not know if the painting's value fluctuated up or down immediately after each sale.

Key concepts

Inequalities in Algebra

Imagine you're at the beach, trying to set up boundaries for a safe swimming area. You plant a flag on one end, where the waves are calm, and another where the waves get rough. Just like the safe zone you’ve marked is between the two flags, an inequality describes a range where a number lies between two given points.

In algebra, inequalities are used to define the bounds of where a variable can land, similar to how you made a safe swimming space. Instead of equality, where x is exactly one number, with inequalities, x could be any number within a range. The compound inequality describes double boundaries, like two flags, showing us where x is 'swimming' safely within two values. So, when Paul Cézanne's artwork fluctuated in value, mathematically we express the range of its price over time using a compound inequality.

Algebraic Expressions

Let's think of algebraic expressions as a secret code. In this code, letters represent numbers, and we use this code to tell stories about how numbers interact. Breaking the code means finding out which numbers the letters stand for.

In our exercise, we have an algebraic expression where 'x' represents the value of the painting. The symbols like \( \backslash leq \) give us clues on how the story unfolds - they tell us that 'x' can be as little as \(252,000, but also it has the potential to soar up to \)28,600,000 over 35 years. These clues are essential in understanding the changes and help us solve the mystery of 'x' throughout the years.

Value of Variables

Consider a treasure map where 'X' marks the spot. In algebra, a variable is that 'X' and finding its value is like discovering where the treasure is hidden. Variables can change; they’re not always buried in the same spot on every map.

In our painting's story, the variable 'x' marks the spot of its value at any time between 1958 and 1993. This 'x' doesn't give us one specific location; it guides us to a vast treasure area. Just like a range of spots on a beach where treasure could be buried, 'x' can be any amount between \(252,000 and \)28,600,000. Each amount is a possible spot where we might dig up the treasure of our painting's value.

Mathematical Reasoning

Mathematical reasoning is the detective work of math. It's how you gather clues, look at the evidence, and solve the mystery. In the question about the painting's worth, you are the detective piecing together the puzzle of its value over time.

By reasoning that the painting sold twice, we can deduce its value changed. It's similar to figuring out how a suspect's story changes over time. Using the evidence of sale prices and understanding the concept of growth, the compound inequality comes together to reveal not a single answer, but an entire story arc of the painting's value. This process is like solving a case, with your badge being mathematical reasoning, helping you conclude the range of worth for the painting between two key points in history.