Step 1a

Given equation is \$3"x"^2 − 14"x" + 8 = 0\$

To solve this equation, we need to factorize the quadratic equation: case - 1

⇒ \$3"x"^2 − 12"x" − 2"x" + 8 = 0\$

⇒ 3x(x − 4) − 2(x − 4)=0

⇒ (3x − 2)(x − 4) = 0

Explanation: Here, the given quadratic equation is factorized into simple from is (3x - 2) (x - 4) = 0.

Step 1b

Setting each factor equal to zero 3x − 2 = 0 and x − 4 = 0.

Explanation: Equating each factor to zero, we distinctively isolate them.

Step 1c

Solving these equations gives us 3x = 2 and x = 4.

\$"x" = 2/3\$ and x = 4

Explanation: Here, the slove for x values are \$"x" = 2/3\$ and x = 4.

Step 2a

To determine continuity, we need to check three conditions:

Explanation: Here, let’s introduce and discuss three conditions.

Step 2b

Let’s evaluate each condition:

1. The function f(x) = 3x - 2 is defined for all real numbers, including x = 4. Therefore, the function is defined at x = 4.

Explanation: Here, use the first condition to satisfy the given function.

Step 2c

2. To find the limit as x approaches 4, we substitute x = 4 into the function: \$ \lim_{x \to 4} 3x - 2 = 3(4) - 2 = 12 - 2 = 10 \$.

Thus, the limit of f(x) as x approaches 4 is 10.

Explanation: Here, use the second condition to satisfy the function.

Step 2d

3. Now, we compare the value of the function at x = 4 with the limit: f(4) = 3(4) - 2 = 12 - 2 = 10.

The value of the function f(x) at x = 4 is equal to the limit.

Since all three conditions are satisfied, we can conclude that the function f(x) = 3x - 2 is continuous at x = 4.

Explanation: Here, satisfying the third condition, we conclude that f(x) = 3x - 2 is continuous at x = 4.