pre cal/trig

Question

1) 6e^(2x-1) + 5 = 15  

2) 12 * 6^(x+4) - 7 = 17  

3) ln (x-2) + ln (x+2) = 6  

4)2|x-5| - 4 = 11  

5) 2|x-4| +9 ≤ 25  

6) 2(x-9)^2 - 6 = 3  

7) x^3 + 3x^2 - 6x - 8 = 0, given x=2 is a solution  

8) -2x^2 + 5x + 6 < 0

Answer 1

1).

The equation is 6e^{(2x - 1)} + 5 = 15.

Subtract 5 from each side.

6e^{(2x - 1)} = 10

Divide each side by 6.

e^{(2x - 1)} = 5/3

e^{(2x - 1)} = 1.66

According to basic exponential rule.

2x - 1 = ln(1.66)

2x - 1 = 0.5068

2x = 0.5068 + 1 = 1.5068

⇒ x = 1.5068/2 = 0.7534.

The solution is x = 0.7534.

2).

The equation is 12 * 6^{(x + 4)} - 7 = 17.

Add 7 to each side.

12 * 6^{(x + 4)}  = 24

Divide each side by 12.

6^{(x + 4)}  = 2

Apply natural log on each side.

ln [6^{(x + 4)} ] = ln 2

Apply power rule : ln [xⁿ] = n ln x.

(x + 4) ln 6 = ln 2

x + 4 = ln 2/ln 6 = 0.3868

x = 0.3868 - 4

⇒ x = - 3.6131.

The solution is x = - 3.6131.

3).

The logarithmic equation is ln (x - 2) + ln (x + 2) = 6.

Apply product property : ln a + ln b = ln (ab).

ln [(x - 2)(x + 2)] = 6

ln (x² - 4) = 6

According to basic rule of logarithm.

x² - 4 = e⁶

x² = 403.4 + 4

x² = 407.42

⇒ x = 20.18.

The solution is x = 20.18.

4).

The absolute value equation is 2| x - 5 | - 4 = 11.

Add 4 to each side.

2| x - 5 | = 15

Divide each side by 2.

| x - 5 | = 15/2

x - 5 = 15/2  and  x - 5 = - 15/2

Add 5 to each side.

x = 15/2 + 5  and  x = - 15/2 + 5

x = (15 + 10)/2  and  x = (- 15 + 10)/2

x = 25/2  and  x = - 5/2.

The solutions are x = 25/2 and x = - 5/2.

Answer 2

Contd.......

5).

The absolute value inequality is 2| x - 4 | + 9 ≤ 25.

Subtract 9 from each side

2| x - 4 | ≤ 16

Divide each side by 2.

| x - 4 | ≤ 8

Case 1 : (x - 4) ≤ 8.

Add 4 to each side.
x ≤ 12.

Case 2 : (x - 4) ≥ - 8.

Add 4 to each side.

x ≥ - 4.

The solution of the inequality is { x | x : x ≤ 12 or x ≥ - 4}.

6).

The equation is 2(x - 9)² - 6 = 3.

Add 6 to each side.

2(x - 9)² = 9

Divide each side by 2.

(x - 9)² = 9/2

Extract square roots from each side.

x - 9 = - 3/√2  and  x - 9 = 3/√2

Add 9 to each side.

x = - 3/√2 + 9  and  x = 3/√2 + 9

x = (- 3 + 9√2)/√2  and  x = (3 + 9√2)/√2.

The solutions are x = (- 3 + 9√2)/√2  and  x = (3 + 9√2)/√2.

7).

The polynomial is x³ + 3x² - 6x - 8 = 0.

x = 2 is a zero of the given polynomial.

By ratiaonal root theorem,

If p/q is a rational zero, then p is a factor of 8 and q is a factor of 1.

The possible values of p are   ± 1, ± 2, and ± 4.

The possible values for q are ± 1.

So, p/q = ± 1, ± 2, and ± 4.

Make a table for the synthetic division and test possible  zeros.

p/q	1	3	- 6	- 8
2	1	5	4	0

Since f(2) = 0, x = 2 is a zero. The depressed polynomial is x² + 5x + 4 = 0.

Since the depressed polynomial of this zero, x² + 5x + 4, is quadratic, use factor by grouping method to find the roots of the related quadratic equation.

x² + 4x + x + 4 = 0

x(x + 4) + 1(x + 4) = 0

Factor : (x + 4)(x + 1) = 0

Apply zero product property.

x + 4 = 0  and   x + 1 = 0

x = - 4  and   x = - 1.

The other zeros of the given polynomial are x = - 1 and x = - 4.

Answer 3

Contd..........

8).

The quadratic inequality is - 2x² + 5x + 6 < 0.

Related equation is - 2x² + 5x + 6 = 0.

- 2x² + 5x + 6, is a quadratic, use quadratic fromula to find the roots of the related quadratic equation.

x = [- b ± √(b² - 4ac)]/2a.

Compare the equation with standard form of quadratic equation : ax² + bx + c = 0

a = - 2, b = 5, and c = 6.

x = [- 5 ± √(5² - 4*-2*6)]/2*-2

x = [- 5 ± √(25 + 48)]/- 4

x = [- 5 ± √73]/- 4

x = [5 + √73]/4  and  x = [5 - √73]/4

x = 3.386  and  x = - 0.886.

(x - 3.386)(x - (- 0.886) < 0

Now, there are two ways this product could be less than zero.One factor must be negative and one must be positive.

First situation:

(x - 3.386) < 0 and (x - (- 0.886) > 0

x < 3.386 and x > - 0.886.

The solution is - 0.866 < x < 3.386.

Second situation:

(x - 3.386) > 0 and (x - (- 0.886) < 0

x > 3.386 and x < - 0.886.

There are NO values for which this situation is true.

The solution of the inequality is {x | - 0.866 < x < 3.386}.