Chapter 5.3, Problem 22PPS

To determine

To solve the inequality:

$\frac{3}{4}x-9\ge 42$

And check the solution.

Answer to Problem 22PPS

$x\ge 68$

Explanation of Solution

Given:

$\frac{3}{4}x-9\ge 42$

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, three fourths of a number x is decreased by 9 can be written as:

$\frac{3}{4}x-9$

Now, then (3/4) x − 9 value is at least is forty two.

  $\begin{array}{l}\frac{3}{4}x-9\ge 42\\ \Rightarrow \frac{3}{4}x-9+9\ge 42+9\\ \Rightarrow \frac{3}{4}x\ge 51\end{array}$ x

Multiply each side by 4/3 to solve the value of x:

  $\begin{array}{c}\frac{3}{4}x\ge 51\\ \frac{4}{3}\cdot \frac{3}{4}x\ge \frac{4}{3}\cdot 51\\ x\ge 68\end{array}$

Thus, the inequality is true for xis greater than 6, thus the solution of given inequality in interval form is:

$\left[68,\infty \right)$

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 68 in left hand side:

$\begin{array}{c}\frac{3}{4}\left(68\right)-9=42\\ =\text{Right Hand}\end{array}$

Now put any value any value of x greater than $68$ , say $72$ and check whether it satisfies the inequality, as

$\begin{array}{l}\frac{3}{4}\left(72\right)-9\ge 42\\ 54-9\ge 42\\ 45\ge 42\end{array}$

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