Math  /  Algebra

QuestionSolve the following inequality algebraically. x5>10|x-5|>10

Studdy Solution

STEP 1

1. The inequality involves an absolute value expression.
2. We need to consider the definition of absolute value to solve the inequality.

STEP 2

1. Understand the absolute value inequality.
2. Split the inequality into two separate inequalities.
3. Solve each inequality for x x .
4. Combine the solutions to express the final solution set.

STEP 3

The absolute value inequality x5>10 |x - 5| > 10 means that the expression inside the absolute value, x5 x - 5 , is more than 10 units away from zero. This can be split into two separate inequalities:
x5>10 x - 5 > 10 x5<10 x - 5 < -10

STEP 4

Solve the first inequality x5>10 x - 5 > 10 :
Add 5 to both sides:
x5>10 x - 5 > 10 x>10+5 x > 10 + 5 x>15 x > 15

STEP 5

Solve the second inequality x5<10 x - 5 < -10 :
Add 5 to both sides:
x5<10 x - 5 < -10 x<10+5 x < -10 + 5 x<5 x < -5

STEP 6

Combine the solutions from the two inequalities. The solution to the inequality x5>10 |x - 5| > 10 is:
x>15orx<5 x > 15 \quad \text{or} \quad x < -5
This means that x x is either greater than 15 or less than -5.
The solution set is:
x(,5)(15,) x \in (-\infty, -5) \cup (15, \infty)

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