Question

solve algebraically. show how to check by sustituting back into original equation.

a) |2x+6|=20  b) |x²+10x+28|=12

Answer 1

b).

The absolute value equation is |x² + 10x + 28| = 12.

Case 1 : x² + 10x + 28 = 12

x² + 10x + 28 - 12 = 0
x² + 10x + 16 = 0

By factor by grouping.

x² + 8x + 2x + 16 = 0

x(x + 8) + 2(x + 8) = 0

(x + 8)(x + 2) = 0

Apply zero product property.

x + 8 = 0  and  x + 2 = 0

x = - 8  and  x = - 2.

Case 2 : - (x² + 10x + 28) = 12

Divide each side by negative 1.

x² + 10x + 28 = - 12

x² + 10x + 28 + 12 = 0
x² + 10x + 40 = 0

x² + 10x + 40 = 0, is a quadratic, use quadratic formula to find the roots of the related equation.

Compare the above equation with ax² + bx + c = 0

a = 1, b = 10, and c = 40.

Solution x = [- b ± sqrt(b² - 4ac)] / 2a.

x = [- 10 ± sqrt(10² - 4*1*40)] / 2*1

x = [- 10 ± sqrt(100 - 160)] / 2

x = [- 10 ± sqrt(- 60)] / 2

x = [- 10 ± 2sqrt(15i²)] / 2

x = [- 5 ± isqrt(15)]

x = [- 5 - isqrt(15)]  and  x = [- 5 + isqrt(15)]

The real solutions of the given absolute value equation are x = - 8 and x = - 2.

Check :

Substitute the value x = - 8 in |x² + 10x + 28| = 12.

|(- 8)² + 10(- 8) + 28| = 12

|64 - 80 + 28| = 12

|12| = 12

12 = 12.

The above statement is true.

So, x = - 8 is a solution of |x² + 10x + 28| = 12.

 

Substitute the value x = - 2 in |x² + 10x + 28| = 12.

|(- 2)² + 10(- 2) + 28| = 12

|4 - 20 + 28| = 12

|12| = 12

12 = 12.

The above statement is true.

So, x = - 2 is a solution of |x² + 10x + 28| = 12.

Answer 2

a).

The absolute value equation is |2x + 6| = 20.

The above equation in the form of |ax + b| = c where c  ≥ 0 and it is equivalent to the statement  ax + b = c or - (ax + b) = c.

Case 1 : 2x + 6 = 20

2x = 20 - 6

2x = 14

x = 14/2

x = 7.

Case 2 : - (2x + 6) = 20

2x + 6 = - 20

2x = - 20 - 6

2x = - 26

x = - 26/2

x = - 13.

The solutions of the absolute value equation are x = - 13 and x = 7.

Check :

Substitute the value x = - 13 in |2x + 6| = 20.

|2(- 13) + 6| = 20

|- 26 + 6| = 20

|- 20| = 20

20 = 20.

The above statement is true.

So, x = - 13 is a solution of |2x + 6| = 20.

 

Substitute the value x = 7 in |2x + 6| = 20.

|2(7) + 6| = 20

|14 + 6| = 20

|20| = 20

20 = 20.

The above statement is true.

So, x = 7 is a solution of |2x + 6| = 20.