We are asked to solve four linear equations for $x$. a) $5x + 4 = -2x - 24$ b) $-3x + 2 = -10x - 5$ c) $2(3x - 4) = -5(2x - 7)$ d) $-4 - 3(5x - 6) = -4(-x - 1) + 6$

1. Problem Description

We are asked to solve four linear equations for $x$.

a) $5x + 4 = -2x - 24$

b) $-3x + 2 = -10x - 5$

c) $2(3x - 4) = -5(2x - 7)$

d) $-4 - 3(5x - 6) = -4(-x - 1) + 6$

2. Solution Steps

a) $5x + 4 = -2x - 24$

Add $2x$ to both sides:

$5x + 2x + 4 = -2x + 2x - 24$

$7x + 4 = -24$

Subtract 4 from both sides:

$7x + 4 - 4 = -24 - 4$

$7x = -28$

Divide both sides by 7:

$\frac{7x}{7} = \frac{-28}{7}$

$x = -4$

b) $-3x + 2 = -10x - 5$

Add $10x$ to both sides:

$-3x + 10x + 2 = -10x + 10x - 5$

$7x + 2 = -5$

Subtract 2 from both sides:

$7x + 2 - 2 = -5 - 2$

$7x = -7$

Divide both sides by 7:

$\frac{7x}{7} = \frac{-7}{7}$

$x = -1$

c) $2(3x - 4) = -5(2x - 7)$

Distribute the constants:

$6x - 8 = -10x + 35$

Add $10x$ to both sides:

$6x + 10x - 8 = -10x + 10x + 35$

$16x - 8 = 35$

Add 8 to both sides:

$16x - 8 + 8 = 35 + 8$

$16x = 43$

Divide both sides by 16:

$\frac{16x}{16} = \frac{43}{16}$

$x = \frac{43}{16}$

d) $-4 - 3(5x - 6) = -4(-x - 1) + 6$

Distribute the constants:

$-4 - 15x + 18 = 4x + 4 + 6$

$-15x + 14 = 4x + 10$

Add $15x$ to both sides:

$-15x + 15x + 14 = 4x + 15x + 10$

$14 = 19x + 10$

Subtract 10 from both sides:

$14 - 10 = 19x + 10 - 10$

$4 = 19x$

Divide both sides by 19:

$\frac{4}{19} = \frac{19x}{19}$

$x = \frac{4}{19}$

3. Final Answer

a) $x = -4$

b) $x = -1$

c) $x = \frac{43}{16}$

d) $x = \frac{4}{19}$