Question

Solve each equation or inequality.  Graph the solution on a number line.  If  there is no solution, write "no solution.

 x^2-x-12/x-5 ≤0

Answer 1

The rational 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

Solutions of related equation x  = -3 and x = 4.

• Step - 3

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

Answer 2

Contd...

• 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

(-∞, -3)            x =  -4                 True

(-3, 4)              x = 0                              False

(4, 5)               x = 4.5                True

(5, ∞)              x = 6                               False

Note that the original inequality contains a “≤ ” symbol, We include it into set of solutions at x = -3 and 4

Above statements are true.

x  = -3 and x  = 4 is a solutions of inequality.

The above conclude that the inequality is satisfied for all x - values in (-∞, -3] and [4, 5).

Solution on number line.

 

Solution 

x ≤ -3

4 ≤ x < 5

The interval notation form of inequality is (-∞, -3] and [4, 5).