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Advanced Math Difficulty: Hard

A rectangle has an area of $155$ square inches. The length of the rectangle is $4$ inches less than $7$ times the width of the rectangle. What is the width of the rectangle, in inches?

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Explanation

The correct answer is $5$ . Let $x$ represent the width, in inches, of the rectangle. It's given that the length of the rectangle is $4$ inches less than $7$ times its width, or $7 x minus 4$ inches. The area of a rectangle is equal to its width multiplied by its length. Multiplying the width, $x$ inches, by the length, $7 x minus 4$ inches, yields $x left parenthesis 7 x minus 4 right parenthesis$ square inches. It’s given that the rectangle has an area of $155$ square inches, so it follows that $x (7 x - 4) = 155$ , or $7 x^{2} - 4 x = 155$ . Subtracting $155$ from both sides of this equation yields $7 x^{2} - 4 x - 155 = 0$ . Factoring the left-hand side of this equation yields $(7 x + 31) (x - 5) = 0$ . Applying the zero product property to this equation yields two solutions: $x = - \frac{31}{7}$ and $x equals 5$ . Since $x$ is the rectangle’s width, in inches, which must be positive, the value of $x$ is $5$ . Therefore, the width of the rectangle, in inches, is $5$ .