Examkin

Question

Difficulty: HardInequalities and Absolute Value Equations

What is the sum of all integer values of xx that satisfy the inequality ∣x2−6x+5∣<2x−2|x^2 - 6x + 5| < 2x - 2?

Answer: 15

Answer

The sum of all integer values of xx satisfying the inequality is 1515.
Factoring both sides gives ∣(x−1)(x−5)∣<2(x−1)|(x - 1)(x - 5)| < 2(x - 1). Since the absolute value is non-negative, the right-hand side requires 2x−2>0  ⟹  x>12x - 2 > 0 \implies x > 1. Under x>1x > 1, the term x−1x - 1 is strictly positive, allowing us to simplify to ∣x−5∣<2|x - 5| < 2, which yields 3<x<73 < x < 7. The integer solutions are 4,5,4, 5, and 66, and their sum is 1515.

Step-by-Step Solution

1
Determine the necessary condition for the right-hand side of the inequality.
Because the left-hand side ∣x2−6x+5∣|x^2 - 6x + 5| is non-negative for all real xx, the right-hand side must be strictly positive. Thus, 2x−2>0  ⟹  x>12x - 2 > 0 \implies x > 1.
An absolute value expression cannot be strictly less than a zero or negative quantity.
2
Factor the quadratic expression inside the absolute value and the linear expression on the right.
∣(x−1)(x−5)∣<2(x−1)|(x - 1)(x - 5)| < 2(x - 1).
Factoring reveals a common linear factor (x−1)(x - 1) on both sides.
3
Simplify the inequality using the condition x>1x > 1.
Since x>1x > 1, we know x−1>0x - 1 > 0, so ∣x−1∣=x−1|x - 1| = x - 1. Splitting the product inside the absolute value gives (x−1)∣x−5∣<2(x−1)(x - 1)|x - 5| < 2(x - 1). Dividing both sides by (x−1)(x - 1) yields ∣x−5∣<2|x - 5| < 2.
Dividing an inequality by a strictly positive number preserves the direction of the inequality sign.
4
Solve the simplified absolute value inequality and sum the integer solutions.
∣x−5∣<2  ⟹  −2<x−5<2  ⟹  3<x<7|x - 5| < 2 \implies -2 < x - 5 < 2 \implies 3 < x < 7. The integer solutions strictly within this range are x=4,5,x = 4, 5, and 66. Their sum is 4+5+6=154 + 5 + 6 = 15.
The integers strictly between 33 and 77 are 44, 55, and 66.

Key Concept

Solving Quadratic Absolute Value Inequalities via Domain Constraints and Factoring
Estimated Time:2m 0s
Rate this question

Topics

Inequalities and Absolute Value Equations

More questions from this topic