Question

1. use the inequality (12 geq 6(12x + 2)).

a. apply the distributive property to the right side.
(12 geq square + square)
b. solve the inequality.
(x leq square)

1. use the inequality (24 geq 58 + 5(x - 3.8)).

a. solve the inequality for (x).
b. which graph shows the solution set of the inequality?
a (longleftarrowhspace{-0.5cm}xrightarrow{hspace{3cm}}hspace{-0.5cm}circhspace{-0.5cm}xrightarrow{hspace{1cm}}longleftarrow) (-10) (0) (10)
b (longleftarrowhspace{-0.5cm}xrightarrow{hspace{2cm}}hspace{-0.5cm}circhspace{-0.5cm}xrightarrow{hspace{2cm}}longleftarrow) (-10) (0) (10)
c (longleftarrowhspace{-0.5cm}xrightarrow{hspace{2cm}}hspace{-0.5cm}\bullethspace{-0.5cm}xrightarrow{hspace{2cm}}longleftarrow) (-10) (0) (10)
d (longleftarrowhspace{-0.5cm}xrightarrow{hspace{1cm}}hspace{-0.5cm}\bullethspace{-0.5cm}xrightarrow{hspace{3cm}}longleftarrow) (-10) (0) (10)

1. gina shows the steps she took to find the solution of the inequality below.

(19 - 2(1 - x) < 13)
(19 - 2 + 2x < 13)
(2x < -4)
(x > -2)
a. should gina have reversed the inequality symbol? explain.
b. write the correct solution for the inequality.

Explanation:

Problem 1

Part a

Step1: Apply Distributive Property

The distributive property is \( a(b + c)=ab+ac \). For \( 6(12x + 2) \), we multiply 6 with \( 12x \) and 6 with 2.
\( 6\times12x = 72x \) and \( 6\times2=12 \), so \( 12\geq72x + 12 \)

Step2: (No further step for part a)

Step1: Subtract 12 from both sides

Subtract 12 from both sides of \( 12\geq72x + 12 \).
\( 12- 12\geq72x+12 - 12 \)
\( 0\geq72x \)

Step2: Divide both sides by 72

Divide both sides by 72 (since 72 is positive, inequality sign remains).
\( \frac{0}{72}\geq\frac{72x}{72} \)
\( 0\geq x \) or \( x\leq0 \)

Step1: Apply Distributive Property

For \( 5(x - 3.8) \), using distributive property \( a(b - c)=ab - ac \), we get \( 5x-19 \). So the inequality becomes \( 24\geq58 + 5x-19 \)

Step2: Simplify the right side

Simplify \( 58-19 = 39 \), so \( 24\geq5x + 39 \)

Step3: Subtract 39 from both sides

Subtract 39 from both sides: \( 24-39\geq5x+39 - 39 \)
\( - 15\geq5x \)

Step4: Divide both sides by 5

Divide both sides by 5 (positive, inequality sign remains): \( \frac{-15}{5}\geq\frac{5x}{5} \)
\( - 3\geq x \) or \( x\leq - 3 \)

Answer:

\( 12\geq\boldsymbol{72x}+ \boldsymbol{12} \)

Part b