Can somebody help me solve these problems?

Question

1) 3x² + 4x + 2 ≥ 0 

2) (2x - 1 )/ (5 - x ) ≤ -3 


3) | 2x - 5 | > 9 
 

Answer 1

1) The inequality 3x ² + 4x  + 2 ≥ 0

Solve the inequality by graphing.

Write the two variable inequality y ≤ 3x ² + 4x + 2.

Write the equation y = 3x ² + 4x  + 2 and it represents a parabola curve.

The graph of the inequality y ≤ 3x ² + 4x  + 2 is the shaded region, so every point in the shaded region satisfies the inequality.

The graph of the equation y = 3x ² + 4x  + 2 is the boundary of the region. Since the inequality symbol is ≤, the boundary is drawn as a solid curve to show that points on the curve doesnot satisfy the inequality.

To graph the boundary curve make the table of values to find ordered pairs that satisfy the equation.

Choose random values for x  and find the corresponding values for y .

x	y = 3x 2 + 4x  + 2	(x, y )
- 1	y = 3(-1)2 + 4(-1) + 2 = 1	(- 1, 1)
- 2	y = 3(-2)2 + 4(-2) + 2 = 6	(- 2, 6)
0	y = 3(0)2 + 4(0) + 2 = 2	(0, 2)
1	y = 3(1)2 + 4(1) + 2 = 9	(1, 9)

To draw inequality y ≤ 3x ² + 4x  + 2 follow the steps.

1.  Draw a coordinate plane.

2.  Plot the points and draw a smooth curve through these points.

3.  To determine which side (out side or in side) to be shaded, use a test point inside the parabola. A simple choice is (-1, 2).

Substitute the value of (x, y) = (-1, 2) in the original inequality.

2 ≤ 3(- 1)² + 4(- 1) + 2

2 < 3 - 4 + 2

2 <  1.

4.  Since the above statement is false, shade the region outside the parabola.

Answer 2

2) The inequality is

• Step-1

State the exclude values,These are the values for which denominator is zero.

The exclude value of the inequality is 5.

• Step - 2

Solve the related equation

Solution of related equation x   = 14.

• Step - 3

Draw the vertical lines at the exclude value and at the solution to separate the number line into intervals.

• Step - 4

Now test  sample values in each interval to determine whether values in the interval satisfy the inequality.

Test interval   x - value      Inequality                           Conclusion

(-∞, 5)              x =  0               False

(5, 14)              x = 6                 True

(14, ∞)            x = 15            False

Note that the original inequality contains a “≤ ” symbol, We include it into set of solutions at x = 14

The above conclude that the inequality is satisfied for all x - values in (5, 14].

Observe the graph the open circle means 5 is not include in the solution set and closed circle means 14 is  include in the solution set .

Solution of the inequality .

5 < x ≤ 14

Answer 3

3) The absolute inequality | 2x  - 5| > 9

Apply the formula |x | > a  is equivalent to x  > a or x  < - a .

Case 1 : 2x  - 5 > 9

2x  > 9 + 5

2x  > 14

x  > 14/2

x  > 7

Case 2 : 2x  - 5 < - 9

2x  < - 9 + 5

2x  < - 4

x  < - 4/2

x  < - 2

The solution set is {x | x  < - 2 or x  > 7}.

Graph the solution set for each inequality and find their intersection.

Observe the graph, the open circle means that - 2 and 7 does not solutions of the inequality.