Leona found three consecutive integers such that the product of $5$ and the sum of the first two was $7$ greater than the opposite of the third. What were her integers?

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Q. Leona found three consecutive integers such that the product of

5

and the sum of the first two was

7

greater than the opposite of the third. What were her integers?

Set Initial Integers: Let $x$ be the first integer. The three consecutive integers can be represented as $x$ , $x+1$ , and $x+2$ .
Calculate Product of Sum: The product of $5$ and the sum of the first two integers is $5 \times (x + (x + 1))$ .
Find Opposite of Third: The opposite of the third integer is $-1 \times (x + 2)$ .
Formulate Equation: According to the problem, $5$ times the sum of the first two integers is $7$ greater than the opposite of the third. This gives us the equation $5 \times (x + (x + 1)) = -1 \times (x + 2) + 7$ .
Simplify Equation: Simplify the equation: $5 \times (2x + 1) = -x - 2 + 7$ .
Distribute and Combine Terms: Distribute the $5$ on the left side of the equation: $10x + 5 = -x - 2 + 7$ .
Isolate Variable Term: Combine like terms on the right side of the equation: $10x + 5 = -x + 5$ .
Solve for x: Add $x$ to both sides to isolate the variable term on one side: $10x + x + 5 = -x + x + 5$ .
Substitute x Values: Simplify the equation: $11x + 5 = 5$ .
Final Consecutive Integers: Subtract $5$ from both sides to solve for $x$ : $11x + 5 - 5 = 5 - 5$ .
Final Consecutive Integers: Subtract $5$ from both sides to solve for $x$ : $11x + 5 - 5 = 5 - 5$ .Simplify the equation: $11x = 0$ .
Final Consecutive Integers: Subtract $5$ from both sides to solve for $x$ : $11x + 5 - 5 = 5 - 5$ . Simplify the equation: $11x = 0$ . Divide both sides by $11$ to solve for $x$ : rac{11x}{11} = rac{0}{11}.
Final Consecutive Integers: Subtract $5$ from both sides to solve for $x$ : $11x + 5 - 5 = 5 - 5$ . Simplify the equation: $11x = 0$ . Divide both sides by $11$ to solve for $x$ : $\frac{11x}{11} = \frac{0}{11}$ . Simplify the equation: $x = 0$ .
Final Consecutive Integers: Subtract $5$ from both sides to solve for $x$ : $11x + 5 - 5 = 5 - 5$ . Simplify the equation: $11x = 0$ . Divide both sides by $11$ to solve for $x$ : $\frac{11x}{11} = \frac{0}{11}$ . Simplify the equation: $x = 0$ . Now that we have the value of $x$ , we can find the three consecutive integers: $x$ , $x$ $0$ , and $x$ $1$ , which are $x$ $2$ , $x$ $3$ , and $x$ $4$ respectively.