Chapter 5.4, Problem 61SR

To determine

To solve the inequality:

  $\frac{2}{3}x+6\ge 22$

And check the solution.

Answer to Problem 61SR

  $x\ge 24$

Explanation of Solution

Given:

  $\frac{2}{3}x+6\ge 22$

Concept Used:

• To check the answer solution for the given inequality
1. First check the solution at end point by converting it into an equation.
2. If the solution is $x<m$, then pick any value of x less than m and check whether it satisfies the inequality.
3. If the solution is $x>m$, then pick any value of x greater than m and check whether it satisfies the inequality.

Properties of inequality:

1. $b\le c\Rightarrow ab\le ac,\text{ if }a\ge 0$
2. $b\le c\Rightarrow ab\ge ac,\text{ if }a\le 0$

Calculation:

Let the number be x, two third of a number x is added to 6 can be written as:

  $\frac{2}{3}x+6$

Now, then (3/4) x − 9 value is at least is twenty two is written as:

  $\begin{array}{l}\frac{2}{3}x+6\ge 22\\ \Rightarrow \frac{2}{3}x+6-6\ge 22-6\\ \Rightarrow \frac{2}{3}x\ge 16\end{array}$

Multiply each side by 3/2 to solve the value of x:

  $\begin{array}{c}\frac{2}{3}x\ge 16\\ \frac{3}{2}\cdot \frac{2}{3}x\ge \frac{3}{2}\cdot 16\\ x\ge 24\end{array}$

Thus, the inequality is true for x is greater than 24, thus the solution of given inequality in interval form is:

  $\left[24,\infty \right)$

Now, to check the answer solution for the given inequality first check the solution at end point by converting it into an equation. Putting x equals to 24 in left hand side:

  $\begin{array}{c}x=24:\frac{2}{3}x+6\ge 22\\ \frac{2}{3}\left(24\right)+6\ge 22\\ 22=22\text{}\end{array}$

Now put any value any value of x greater than 24, say $27$ and check whether it satisfies the inequality, as

  $\begin{array}{c}x=27:\frac{2}{3}\left(27\right)+6\ge 22\\ 18+6\ge 22\\ 24\ge 22\text{}\end{array}$

Since, both conditions are satisfied thus solution of inequality is correct.