Solve each of the inequalities and express the solution sets in interval notation. Match each of the inequalities with the interval to obtain a true statement.
-$( - \infty , 18 ]$
A) $\frac { x + 4 } { 22 } - \frac { x + 9 } { 9 } \geq \frac { 7 } { 18 }$
B) $\frac { x - 5 } { 20 } - \frac { x - 2 } { 25 } \leq \frac { 1 } { 100 }$
C) $\frac { 6 x - 5 } { 10 } - \frac { x - 1 } { 20 }

Question

Solve each of the inequalities and express the solution sets in interval notation. Match each of the inequalities with the interval to obtain a true statement.
-$( - \infty , 18 ]$ 
A) $\frac { x + 4 } { 22 } - \frac { x + 9 } { 9 } \geq \frac { 7 } { 18 }$ 
B) $\frac { x - 5 } { 20 } - \frac { x - 2 } { 25 } \leq \frac { 1 } { 100 }$ 
C) $\frac { 6 x - 5 } { 10 } - \frac { x - 1 } { 20 } < - 4$

Quizplus · Accepted Answer

To solve each inequality, we need to simplify and isolate the variable on one side.

A) $\frac { x + 4 } { 22 } - \frac { x + 9 } { 9 } \geq \frac { 7 } { 18 }$

First, find a common denominator of 198:

$\frac{(x+4) \cdot 9}{198} - \frac{(x+9) \cdot 22}{198} \geq \frac{77}{198}$

Simplify:

$\frac{-13x - 170}{198} \geq \frac{77}{198}$

Isolate the variable:

$-13x - 170 \geq 77$

$-13x \geq 247$

$x \leq -19$

The solution set is $(-\infty, -19]$, which does not match the given interval of $(-\infty, 18]$.

B) $\frac { x - 5 } { 20 } - \frac { x - 2 } { 25 } \leq \frac { 1 } { 100 }$

Find a common denominator of 500:

$\frac{25(x-5)}{500} - \frac{20(x-2)}{500} \leq \frac{1}{100}$

Simplify:

$\frac{5x - 133}{500} \leq \frac{1}{100}$

Isolate the variable:

$5x - 133 \leq 5$

$5x \leq 138$

$x \leq \frac{138}{5}$

The solution set is $(-\infty, \frac{138}{5}]$, which matches the given interval of $(-\infty, 18]$.

C) $\frac { 6 x - 5 } { 10 } - \frac { x - 1 } { 20 } < - 4$

Find a common denominator of 20:

$\frac{12x-10-x+1}{20} < -4$

Simplify:

$\frac{11x-9}{20} < -4$

Isolate the variable:

$11x - 9 < -80$

$11x < -71$

$x < -\frac{71}{11}$

The solution set is $(-\infty, -\frac{71}{11})$, which does not match the given interval of $(-\infty, 18]$.

Therefore, the best choice is B, which matches the given interval.

Solve Each of the Inequalities and Express the Solution Sets (−∞,18]( - \infty , 18 ](−∞,18]

Solve Each of the Inequalities and Express the Solution Sets $( - \infty , 18 ]$