Chapter 8.8, Problem 57HP

(a)

To determine

Identify the equation have no solution or infinitely many solution.

Answer to Problem 57HP

Infinitely Many Solution.

Explanation of Solution

Given:

The inequality: $5x-6\ge 3(x-2)+2x$

Concept Used:

If an equation is x = a, this means the equation is true only when the variable assumes the value a. The equation has only one solution exists.

If an equation a = a, this means the equation is true for any value of the variable, has infinitely many solution.

If a = b, this means there is no value of the variable that will make a equal to b , has no solution.

Calculation:

The inequality: $5x-6\ge 3(x-2)+2x$

  $\begin{array}{l}5x-6\ge 3(x-2)+2x\\ 5x-6\ge 3x-6+2x[\text{Distribute with 3 ]}\\ 5x-6\ge 5x-6[\text{Simplify}]\\ 5x-5x-6\ge 5x-5x-6\\ -6\ge -6[\text{True statement ]}\end{array}$infinitely many solutions Once again, the x terms cancel out, but we are left with a true statement. - 6 does equal - 6, and will no matter what we put in for x.

Therefore we can conclude that there is infinitely many solutions. Whatever number we put in for the variable x, it'll always give you a true statement.

Thus, this inequality is having infinitely many solutions; the inequality is true for all real values of x.

(b)

To determine

Identify the equation have no solution or infinitely many solution.

Answer to Problem 57HP

No Solution.

Explanation of Solution

Given:

The inequality: $12p+17\le 3(4p-8)$

Concept Used:

If an equation is x = a, this means the equation is true only when the variable assumes the value a. The equation has only one solution exists.

If an equation a = a, this means the equation is true for any value of the variable, has infinitely many solution.

If a = b, this means there is no value of the variable that will make a equal to b , has no solution.

Calculation:

The inequality: $12p+17\le 3(4p-8)$

  $\begin{array}{l}12p+17\le 3(4p-8)\\ 12p+17\le 12p-24\\ 12p-12p+17\le 12p-12-24\\ 17\le -24[\text{False statement ]}\end{array}$

No Solution If the coefficients of variable are the same on both sides are equal, but the constants are different, then no solutions will occur.

If the equation ends with a false statement (example: a = b) then you know that there`s no solution.

Thus, this inequality is having no solution.