Question

solve each inequality

1. $x - 17 < -6$

$\underline{+17\\ \\ \\ \\ +17}$
$x < 11$
(number line plot)

1. $-\frac{y}{6} > 8$

$y < -48$
(number line plot)

1. $\frac{y}{4} - 7 > 5$

$\underline{+7\\ \\ \\ +7}$
$\frac{y}{4} > \frac{12}{4}$
$y > 12$
(number line plot)

1. $-5 \leq -\frac{4}{9}b + 7$

$\underline{-7\\ \\ \\ \\ \\ \\ \\ -7}$
$-12 \leq -\frac{4}{9}b$
multiply both sides by $-\frac{9}{4}$ (reverse inequality):
$27 \geq b$
(number line plot)

1. $2x - 12 > 5x - 4$

$2x - 5x > -4 + 12$
$-3x > 8$
$x < -\frac{8}{3} = -2\frac{2}{3}$
(number line plot)

1. $-3(5x - 8) > 5 - 7x + 2$

$-15x + 24 > 5 - 7x + 2$
$-15x + 24 > 7 - 7x$
$-15x + 7x > 7 - 24$
$-8x > -17$
divide by $-8$ (reverse inequality):
$x < \frac{17}{8} = 2\frac{1}{8}$
(number line plot)

1. (incomplete, shows division: $\begin{array}{r}5 \overline{)117}

-10\\ \\ \\ \\
\hline 17
-15
\hline 2end{array}$)

Explanation:

Let's solve the first inequality \( x - 17 < -6 \) step by step:

Step 1: Isolate the variable \( x \)

To solve for \( x \), we need to get rid of the \(-17\) on the left side. We can do this by adding \( 17 \) to both sides of the inequality. This is based on the addition property of inequalities, which states that if we add the same number to both sides of an inequality, the direction of the inequality sign remains the same.

So, we have:
\( x - 17 + 17 < -6 + 17 \)

Step 2: Simplify both sides

Simplifying the left side: \( x - 17 + 17 = x \)

Simplifying the right side: \( -6 + 17 = 11 \)

Putting it together, we get:
\( x < 11 \)

Answer:

\( x < 11 \)