Solve for c.

Question

2c–11
3
≥3 or
3c–2
4
<4
Write your answer as a compound inequality with integers.
or

Answer 1

To solve for $c$ in the inequalities:

1. $\frac{2c-11}{3}\ge 3$
2. $\frac{3c-2}{4}<4$

we'll solve each inequality separately.

For the first inequality:

$$\frac{2c-11}{3}\ge 3$$

Multiply both sides by 3 (to eliminate the fraction):

$$2c-11\ge 9$$

Add 11 to both sides:

$$2c\ge 20$$

Divide by 2:

$$c\ge 10$$

For the second inequality:

$$\frac{3c-2}{4}<4$$

Multiply both sides by 4:

$$3c-2<16$$

Add 2 to both sides:

$$3c<18$$

Divide by 3:

$$c<6$$

Combine the results:

We have two inequalities:

1. $c\ge 10$
2. $c<6$

When combined as a compound inequality, since these two inequalities do not overlap (i.e., there are no values of $c$ that satisfy both conditions at the same time), we write the solution in terms of $c$:

Final Answer:

$c\ge 10$ or $c<6$

As a compound inequality with integers, the solution can be represented as:

$$c\in (-\mathrm{\infty },6)\cup [10,\mathrm{\infty })$$

In terms of integer solutions explicitly:

$$c<6\text{or}c\ge 10$$

This is the complete solution for $c$.