Question

Given the inequality |x-1| < 5, which of the following inequalities is equivalent to it?

a) x - 11 < 5
b) x + 1 < 5
c) x - 6 < 0
d) x + 6 > 1

Answer 1

The equivalent inequality is c) x - 6 < 0. Option C is correct.

Step-by-step explanation:

The given inequality is |x - 1| < 5. To find the equivalent inequality, we consider two cases:

Case 1: When (x - 1) is positive or zero, the inequality becomes (x - 1) < 5.

Adding 1 to both sides, we get x < 6.

Case 2: When (x - 1) is negative, the inequality becomes -(x - 1) < 5.

Multiplying both sides by -1 and adding 1, we get x > -4.

Combining the results from both cases, we have -4 < x < 6. However, to find the equivalent inequality in the given options, we need to express this in the form of x - k < 0.

Subtracting 5 from all parts, we get -9 < x - 5 < 1. Now, subtracting 1 from all parts, we get -10 < x - 6 < 0.

Thus, the equivalent inequality is x - 6 < 0, which matches option c).

In summary, by considering both cases and manipulating the inequality, we find that x - 6 < 0 is the equivalent form, aligning with option c).