Question

1/3n - 2 1/2 > -5
please do a step by step

Answer 1

`n < -(2)/(21)` or
`n > 0`.

Explanation:

1. Write the expression.

`(1)/(3n) -2+(1)/(2) > -5`

From this point forward, we're not solving the inequation as we may solve a normal one. Since it's a rational inequation (the variable is dividing another number), let's write it in standard form.

Standard form. The inequality is expressed by only one fraction or term, compared to 0.

2. Add 5 to both sides of the equation.

`(1)/(3n) -2+(1)/(2)+5 > -5+5\\ \\(1)/(3n) -2+(1)/(2)+5 > 0`

3. Sum all the constants.

`(1)/(3n) +(-2+(1)/(2)+5) > 0\\ \\(1)/(3n)+ (-(4)/(2) +(1)/(2)+(10)/(2) ) > 0\\ \\(1)/(3n)+ ((-4+1+10)/(2) ) > 0\\ \\(1)/(3n) +((7)/(2) ) > 0`

4. Sum all the expressions.

`((1)(2)+(3n)(7))/((3n)(2)) > 0\\ \\(2+21n)/(6n) > 0`

5. Solve the numerator as an equation.

`2+21n=0\\ \\21n=-2\\ \\n=(-2)/(21)`

6. Solve the denominator as an equation.

`6n=0\\ \\n=(0)/(6) \\ \\n=0`

7. Graph the values on the number line.

Check attached image 1.

8. Test the original expression with values from the regions of interest.

a. First, let's test with a value that goes from -∞ to -2/21 (Region 1).

`(1)/(3((-2)/(22) )) -2+(1)/(2) > -5\\ \\-5.167 > -5`

The statement is true, therefore, this region is one of the solutions. Hence, one of the solutions is
`n < -(2)/(21)`

b. Test with a value that goes from -2/21 to 0 (Region 2).

`(1)/(3((-2)/(10) )) -2+(1)/(2) > -5\\ \\-3.167 > -5`

The statement is false, therefore, the region is not one of the solutions.

c. First, let's test with a value that goes from 0 to ∞ (Region 3).

`(1)/(3(1)) -2+(1)/(2) > -5\\ \\-1.166 > -5`

The statement is true, therefore, this region is one of the solutions. Hence, the second solution is
`n > 0`

9. Express your results.

`n < -(2)/(21)` or
`n > 0`.