1. The problem involves evaluating inequalities and expressions given as: - $x7 - 4$ - $\frac{1}{4}x$ - $3$ - $2c = -4$ - $2c < -4$ - $24c > -8$ 2. First, interpret each expression: - $x7 - 4$ likely means $7x - 4$. - $\frac{1}{4}x$ is a linear expression. - $3$ is a constant. - $2c = -4$ is an equation. - $2c < -4$ is an inequality. - $24c > -8$ is another inequality. 3. Solve the equation $2c = -4$: $$ 2c = -4 \\ c = \frac{-4}{2} = -2 $$ 4. Solve the inequality $2c < -4$: $$ 2c < -4 \\ c < \frac{-4}{2} = -2 $$ 5. Solve the inequality $24c > -8$: $$ 24c > -8 \\ c > \frac{-8}{24} = -\frac{1}{3} $$ 6. Check if these solutions exist: - The equation $2c = -4$ has a solution $c = -2$. - The inequality $2c < -4$ has solutions for all $c < -2$. - The inequality $24c > -8$ has solutions for all $c > -\frac{1}{3}$. 7. Note that the inequalities $2c < -4$ and $24c > -8$ do not overlap because $c$ cannot be simultaneously less than $-2$ and greater than $-\frac{1}{3}$. 8. The expressions $7x - 4$, $\frac{1}{4}x$, and $3$ are linear or constant and exist for all real $x$. Final answers: - $c = -2$ satisfies $2c = -4$. - $c < -2$ satisfies $2c < -4$. - $c > -\frac{1}{3}$ satisfies $24c > -8$. - Expressions $7x - 4$, $\frac{1}{4}x$, and $3$ exist for all real $x$.