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Solve Compound Inequality for Vehicle Miles Driven from Depreciation Formula
Mathematics
Grade 9 of Junior High School
Question Content
The value (V, in dollars) of a vehicle depends on the miles (x) that it has been driven, as given by the formula $V = 20,000 - 0.4x$. After one year, the value of the vehicle is between $18,000 and $19,000, which can be expressed as follows: $18,000 \\leq 20,000 - 0.4x \\leq 19,000$. Which compound inequality represents the range for the miles driven? Answer choices are rounded to the nearest whole number. Answer options: $400 \\leq x \\leq 800$; $4,000 \\leq x \\leq 8,000$; $15,200 \\leq x \\leq 15,600$; $2,500 \\leq x \\leq 5,000$
Correct Answer
$2,500 \\leq x \\leq 5,000$
Detailed Solution Steps
1
Step 1: Start with the given compound inequality for the vehicle value: $18,000 \\leq 20,000 - 0.4x \\leq 19,000$
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Step 2: Subtract 20,000 from all three parts of the inequality to isolate the term with $x$: $18,000 - 20,000 \\leq 20,000 - 0.4x - 20,000 \\leq 19,000 - 20,000$, which simplifies to $-2,000 \\leq -0.4x \\leq -1,000$
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Step 3: Divide all three parts by -0.4. Remember that dividing an inequality by a negative number reverses the inequality symbols: $\\frac{-2,000}{-0.4} \\geq x \\geq \\frac{-1,000}{-0.4}$
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Step 4: Calculate the values: $5,000 \\geq x \\geq 2,500$, which can be rewritten as $2,500 \\leq x \\leq 5,000$
Knowledge Points Involved
1
Compound Inequalities
A compound inequality combines two or more inequalities into one statement, representing a range of values that satisfy both conditions. They are often written in the form $a \\leq x \\leq b$ to show values of $x$ that are between $a$ and $b$. They are used to model real-world scenarios where a quantity has an upper and lower limit, like the value of a depreciating vehicle here.
2
Inequality Properties with Negative Coefficients
When multiplying or dividing all parts of an inequality by a negative number, the direction of the inequality symbols ($\\leq$, $\\geq$, $<$, $>$) must be reversed. This rule ensures the inequality remains true, as multiplying/dividing by a negative number flips the relative size of the values being compared.
3
Solving Linear Inequalities for a Variable
To solve a linear inequality for a variable, use inverse operations (subtraction, addition, multiplication, division) to isolate the variable on one side of the inequality. The process is similar to solving linear equations, but with the key exception of reversing symbols when using negative numbers in multiplication/division.
4
Real-World Linear Model Applications
Linear equations like $V = 20,000 - 0.4x$ are used to model real-world rate-based changes, such as vehicle depreciation. Here, the equation represents a constant rate of value loss ($0.40 per mile driven), and inequalities derived from it can find valid ranges for the independent variable (miles driven) based on limits for the dependent variable (vehicle value).
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