Question

in exercises 1–6, translate the given statement into an algebraic inequality or compound inequality

1. to avoid an additional charge, the sum of the length, ( l ), width, ( w ), and depth, ( d ), of a piece of luggage to be checked on a commercial airline can be at most 61 inches.

1. a pg-13 movie rating means that your age, ( a ), must be at least 13 years for you to view the movie.

1. the cost, ( c(a) ), of renting a car from company a is less expensive than the cost ( c(b) ) of renting from company b.

1. the label on a bottle of film developer states that the temperature, ( t ), of the contents must be kept between ( 68^circ ) and ( 77^circ )f.

1. you are in a certain tax bracket if your taxable income, ( i ), is over $24,650, but not over $59,750.

1. the range of temperature, ( t ), on the surface of mars is from ( -153^circ )c to ( 20^circ )c.

solve exercises 7–14 graphically and algebraically.

1. ( 3x > -6 )

1. ( 3 - 2x leq 5 )

1. ( x + 2 > 3x - 8 )

1. ( 5x - 1 < 2x + 11 )

Explanation:

Problem 7: Solve \( 3x > -6 \) algebraically and graphically

Algebraic Solution:

Step 1: Isolate \( x \)

To solve for \( x \), divide both sides of the inequality by 3. Since 3 is positive, the direction of the inequality sign remains the same.

$$ \frac{3x}{3} > \frac{-6}{3} $$

Step 2: Simplify

Simplifying both sides gives:

$$ x > -2 $$

Graphical Solution:

Step 1: Draw the number line

Draw a horizontal line with a scale. Mark the point \( -2 \) on the number line.

Step 2: Represent the inequality

Since the inequality is \( x > -2 \), we use an open circle at \( -2 \) (because \( x = -2 \) is not included) and draw an arrow to the right (indicating all numbers greater than \( -2 \)).

Problem 8: Solve \( 3 - 2x \leq 5 \) algebraically and graphically

Algebraic Solution:

Step 1: Subtract 3 from both sides

Subtract 3 from both sides to isolate the term with \( x \):

$$ 3 - 2x - 3 \leq 5 - 3 $$

$$ -2x \leq 2 $$

Step 2: Divide by -2 (and reverse the inequality)

Divide both sides by -2. Since we are dividing by a negative number, the direction of the inequality sign reverses:

$$ \frac{-2x}{-2} \geq \frac{2}{-2} $$

$$ x \geq -1 $$

Graphical Solution:

Answer:

Step 1: Draw the number line

Draw a horizontal line with a scale. Mark the point \( 4 \) on the number line.

Step 2: Represent the inequality

Since the inequality is \( x < 4 \), we use an open circle at \( 4 \) (because \( x = 4 \) is not included) and draw an arrow to the left (indicating all numbers less than \( 4 \)).

Final Answers:

1. Algebraic: \( x > -2 \); Graphical: Open circle at \( -2 \), arrow right.
2. Algebraic: \( x \geq -1 \); Graphical: Closed circle at \( -1 \), arrow right.
3. Algebraic: \( x < 5 \); Graphical: Open circle at \( 5 \), arrow left.
4. Algebraic: \( x < 4 \); Graphical: Open circle at \( 4 \), arrow left.