Question

How do you solve inequalities $5 - 3c \leqslant c + 17$ ?

Answer 1

For the given inequality $5 - 3c \leqslant c + 17$ , we have to solve for $c$ . We just need to remember to apply all of our operations to both the parts, like adding , subtracting , multiplying, and dividing . Signs get reversed only when we multiply or divide each side by a negative number.

Complete step by step solution:
The given inequality is $5 - 3c \leqslant c + 17$ .
We just need to remember to apply all of our operations to both the parts.
Subtract $- 5$ from both the side of the inequality ,
$\Rightarrow 5 - 3c - 5 \leqslant c + 17 - 5$
$\Rightarrow - 3c \leqslant c + 12$
We can also write it as ,
$\Rightarrow c + 12 \geqslant - 3c$
Add $3c$both the side of the inequality ,
$\Rightarrow c + 12 + 3c \geqslant - 3c + 3c$
$\Rightarrow 4c + 12 \geqslant 0$
Subtract $12$ from both the side of the inequality ,
$\Rightarrow 4c + 12 - 12 \geqslant - 12$
$\Rightarrow 4c \geqslant - 12$
Divide both the side of the inequality by $4$,
$\Rightarrow \dfrac{{4c}}{4} \geqslant \dfrac{{ - 12}}{4} \\\ \Rightarrow c \geqslant - 3 \\\$
$\Rightarrow c \geqslant - 3$
Therefore $c \in [ - 3,\infty )$ .

Additional Informational: Inequalities are often manipulated like equations and follow terribly similar rules, however there's one necessary exception. If you add constant variety to each side of given inequality , the inequality remains true. If you subtract constant variety from each side of the given inequality , the inequality remains true. If you multiply or divide each side of the given inequality by constant positive variety, the inequality remains true.

However, if you multiply or divide each side of the given inequality by a negative variety, the inequality isn't any longer true. In fact, the inequality becomes reversed. This can be quite simple to visualize as a result of we will write that $4 > 2$ . However, if we tend to multiply each side of this inequality by a negative number, we've got to reverse the inequality , giving $- 4 < \- 2$ in order for it to be true.

Note: In inequality:
i) You can add constant amount to every aspect
ii) You can subtract constant amount from both sides
iii) You can multiply or divide both sides by a constant positive amount .
iv) If you multiply or divide both sides by a negative amount, the inequality needs to be reversed.