Question

solve each inequality

1. $x - 17 < -6$

$\underline{+17\\ \\ \\ +17}$
$x < 11$
(number line with 0 and 11, arrow left from 11)

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

(work shown, result $y < -49$)
(number line with -49 and 0, arrow left from -49)

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

$\underline{+7\\ \\ \\ +7}$
$\frac{y}{4} > \frac{12}{4}$
$y > 12$? wait, no, $\frac{y}{4} > 12/4$ → $y/4 > 3$ → $y > 12$? but the number line has 0 and 3, arrow right from 3. maybe typo, original is $\frac{y}{4} - 7 > 5$ → $\frac{y}{4} > 12$ → $y > 48$? but the work shows $\frac{y}{4} > \frac{12}{4}$ → $y/4 > 3$ → $y > 12$? wait the number line is 0 to 3, arrow right from 3. maybe the original problem was $\frac{y}{4} - 7 > 5$? no, maybe $\frac{y}{4} - 7 > 5$ → $\frac{y}{4} > 12$ → $y > 48$. but the work has $\frac{y}{4} > \frac{12}{4}$ → $y/4 > 3$ → $y > 12$? maybe the problem was $\frac{y}{4} - 7 > 5$ but miscalculation. anyway, ocr text for problem 3: $\frac{y}{4} - 7 > 5$, $\underline{+7\\ \\ \\ +7}$, $\frac{y}{4} > \frac{12}{4}$, (number line 0 to 3, arrow right from 3)

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 0 to 27, arrow left from 27)

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

$2x - 5x > -4 + 12$
$-3x > 8$
(divide by -3, reverse inequality)
$x < -\frac{8}{3} = -2\frac{2}{3}$
(number line with -3, -2, 0, arrow left from -2 2/3)

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

$-15x + 24 > 5 - 7x + 2$
$-15x + 24 > 7 - 7x$
(subtract 24, add 7x)
$-8x > -17$
(divide by -8, reverse inequality)
$x < \frac{17}{8} = 2\frac{1}{8}$
(number line 0 to 2, arrow left from 2 1/8)

1. (partially visible, work with fractions)

Explanation:

Step1: Solve \( x - 17 < -6 \)

Add 17 to both sides: \( x - 17 + 17 < -6 + 17 \)
Simplify: \( x < 11 \)

Step2: Solve \( -\frac{y}{7} > 7 \) (assuming original was \( -\frac{y}{7} > 7 \))

Multiply both sides by -7 (reverse inequality): \( y < -49 \)

Step3: Solve \( \frac{y}{4} - 7 > 5 \)

Add 7 to both sides: \( \frac{y}{4} > 12 \)
Multiply by 4: \( y > 48 \) (note: original work had a typo, correct is \( y > 48 \), not \( \frac{4a}{4} > \frac{12}{4} \))

Step4: Solve \( -5 \leq -\frac{4}{9}b + 7 \) (assuming \( -\frac{4}{9}b \))

Subtract 7: \( -12 \leq -\frac{4}{9}b \)
Multiply by \( -\frac{9}{4} \) (reverse inequality): \( 27 \geq b \) or \( b \leq 27 \)

Step5: Solve \( 2x - 12 > 5x - 4 \)

Subtract \( 5x \) and add 12: \( -3x > 8 \)
Divide by -3 (reverse inequality): \( x < -\frac{8}{3} \approx -2.67 \)

Step6: Solve \( -3(5x - 8) > 5 - 7x + 2 \)

Distribute: \( -15x + 24 > 7 - 7x \)
Add \( 15x \) and subtract 7: \( 17 > 8x \)
Divide by 8: \( x < \frac{17}{8} = 2\frac{1}{8} \)

Answer:

1. \( x < 11 \)
2. \( y < -49 \)
3. \( y > 48 \) (corrected)
4. \( b \leq 27 \)
5. \( x < -\frac{8}{3} \)
6. \( x < \frac{17}{8} \)