First, solve for \( \left|x - 2\right| \):

\( \left|x - 2\right| \) + 1 = 9
\( \left|x - 2\right| \) = 9 - 1
\( \left|x - 2\right| \) = 8

The value inside the absolute value brackets can be either positive or negative so (x - 2) must equal + 8 or -8 for \( \left|x - 2\right| \) to equal 8:

So, x = -6 or x = 10.

A number with an exponent (b^e) consists of a base (b) raised to a power (e). The exponent indicates the number of times that the base is multiplied by itself. A base with an exponent of 1 equals the base (b¹ = b) and a base with an exponent of 0 equals 1 ( (b⁰ = 1).

If the tiger has consumed 54 pounds of food in 9 days that's \( \frac{54}{9} \) = 6 pounds of food per day. The tiger needs to consume 78 - 54 = 24 more pounds of food to reach 78 pounds total. At 6 pounds of food per day that's \( \frac{24}{6} \) = 4 more days.

An integer is any whole number, including zero. An integer can be either positive or negative. Examples include -77, -1, 0, 55, 119.

To add these fractions, first find the lowest common multiple of their denominators. The first few multiples of 5 are [5, 10, 15, 20, 25, 30, 35, 40, 45, 50] and the first few multiples of 9 are [9, 18, 27, 36, 45, 54, 63, 72, 81, 90]. The first few multiples they share are [45, 90] making 45 the smallest multiple 5 and 9 share.

Next, convert the fractions so each denominator equals the lowest common multiple:

\( \frac{6 x 9}{5 x 9} \) + \( \frac{2 x 5}{9 x 5} \)

\( \frac{54}{45} \) + \( \frac{10}{45} \)

Now, because the fractions share a common denominator, you can add them:

\( \frac{54 + 10}{45} \) = \( \frac{64}{45} \) = 1\(\frac{19}{45}\)