write the word sentence as an inequality. 1. a number ( x ) is at most 3. 2. a number ( y ) added to 2 is greater than 7. 3. a number ( c ) multiplied by 3 is less than -12. 4. a number ( m ) minus 1.5 is no less than 2. 5. your friend writes the word sentence as an inequality. is your friend correct? explain your reasoning. three times a number ( x ) is more than 18. ( 3x 1 ) 11. ( n leq -2.5 ) 12. in order to try out for one of the parts in a play at the local theater, you must be at most 12 years old. write an inequality that represents this situation. tell whether the given value is a solution of the inequality. 13. ( 3h - 7

Question

write the word sentence as an inequality.

a number ( x ) is at most 3.
a number ( y ) added to 2 is greater than 7.
a number ( c ) multiplied by 3 is less than -12.
a number ( m ) minus 1.5 is no less than 2.
your friend writes the word sentence as an inequality. is your friend correct? explain your reasoning.

three times a number ( x ) is more than 18.
( 3x < 18 )
tell whether the given value is a solution of the inequality.

( t - 3 geq 2, t = 10 )
( 6w < -2, w = 1 )
( p + 1.6 leq 4, p = 5 )
( \frac{3}{2}d > -3, d = 0 )

graph the inequality on a number line.

( k > 1 )
( n leq -2.5 )
in order to try out for one of the parts in a play at the local theater, you must be at most 12 years old. write an inequality that represents this situation.

tell whether the given value is a solution of the inequality.

( 3h - 7 < h, h = 2 )
( q + 8 geq \frac{5}{4}, q = -12 )
consider the inequalities ( -2x < 10 ) and ( -6 < -2x ).

a. is ( x = 0 ) a solution of both inequalities?
b. is ( x = 4 ) a solution of both inequalities?
c. find another value of ( x ) that is a solution of both inequalities.

the maximum area that is available for a rectangular garden is 80 square feet.

a. write an inequality that represents the possible dimensions for the garden.
b. find three different sets of allowable dimensions for the garden. find the area of each garden.

Accepted Answer

Let's solve question 1 first: # Explanation: ## Step1: Understand "at most" "At most" means the number $ x $ is less than or equal to 3. ## Step2: Write the inequality So the inequality is $ x \leq 3 $. # Answer: $ x \leq 3 $ Question 2: # Explanation: ## Step1: Translate "added to" "A number $ y $ added to 2" is $ y + 2 $. ## Step2: Translate "greater than" "Is greater than 7" means $ y + 2 > 7 $. # Answer: $ y + 2 > 7 $ Question 3: # Explanation: ## Step1: Translate "multiplied by" "A number $ c $ multiplied by 3" is $ 3c $. ## Step2: Translate "less than" "Is less than -12" means $ 3c < -12 $. # Answer: $ 3c < -12 $ Question 4: # Explanation: ## Step1: Translate "minus" "A number $ m $ minus 1.5" is $ m - 1.5 $. ## Step2: Translate "no less than" "No less than 2" means greater than or equal to 2, so $ m - 1.5 \geq 2 $. # Answer: $ m - 1.5 \geq 2 $ Question 5: # Explanation: ## Step1: Analyze the word sentence "Three times a number $ x $ is more than 18" means $ 3x > 18 $. ## Step2: Compare with friend's inequality Friend wrote $ 3x < 18 $, which is the opposite. So the friend is incorrect because "more than" means greater than, not less than. # Answer: The friend is incorrect. The correct inequality should be $ 3x > 18 $ (friend wrote $ 3x < 18 $). Question 6: # Explanation: ## Step1: Substitute $ t = 10 $ into $ t - 3 \geq 2 $ Calculate $ 10 - 3 = 7 $. ## Step2: Check the inequality $ 7 \geq 2 $ is true. # Answer: Yes, $ t = 10 $ is a solution. Question 7: (Assuming the inequality is $ 6w < -7 $, $ w = 1 $) # Explanation: ## Step1: Substitute $ w = 1 $ into $ 6w < -7 $ Calculate $ 6(1) = 6 $. ## Step2: Check the inequality $ 6 < -7 $ is false. # Answer: No, $ w = 1 $ is not a solution. Question 8: # Explanation: ## Step1: Substitute $ p = 5 $ into $ p + 1.6 \leq 4 $ Calculate $ 5 + 1.6 = 6.6 $. ## Step2: Check the inequality $ 6.6 \leq 4 $ is false. # Answer: No, $ p = 5 $ is not a solution. Question 9: (Assuming the inequality is $ \frac{3}{2}d > -3 $, $ d = 0 $) # Explanation: ## Step1: Substitute $ d = 0 $ into $ \frac{3}{2}d > -3 $ Calculate $ \frac{3}{2}(0) = 0 $. ## Step2: Check the inequality $ 0 > -3 $ is true. # Answer: Yes, $ d = 0 $ is a solution. Question 10: To graph $ k > 1 $ on a number line: # Explanation: ## Step1: Draw a number line Mark 1 with an open circle (since $ k $ is not equal to 1). ## Step2: Shade the region Shade the part of the number line to the right of 1 (since $ k $ is greater than 1). # Answer: (Graph description: Open circle at 1, shade to the right.) Question 11: To graph $ n \leq -2.5 $ on a number line: # Explanation: ## Step1: Draw a number line Mark -2.5 with a closed circle (since $ n $ can be equal to -2.5). ## Step2: Shade the region Shade the part of the number line to the left of -2.5 (since $ n $ is less than or equal to -2.5). # Answer: (Graph description: Closed circle at -2.5, shade to the left.) Question 12: # Explanation: ## Step1: Define the variable Let $ a $ be your age. ## Step2: Translate "at most 12" "At most" means less than or equal to, so $ a \leq 12 $. # Answer: $ a \leq 12 $ (where $ a $ is your age) Question 13: # Explanation: ## Step1: Substitute $ h = 2 $ into $ 3h - 7 < h $ Calculate $ 3(2) - 7 = 6 - 7 = -1 $, and $ h = 2 $. ## Step2: Check the inequality $ -1 < 2 $ is true. # Answer: Yes, $ h = 2 $ is a solution. Question 14: (Assuming the inequality is $ q + 8 \geq \frac{5}{4} $, $ q = -12 $) # Explanation: ## Step1: Substitute $ q = -12 $ into $ q + 8 \geq \frac{5}{4} $ Calculate $ -12 + 8 = -4 $. ## Step2: Check the inequality $ -4 \geq \frac{5}{4} $ (which is 1.25) is false. # Answer: No, $ q = -12 $ is not a solution. Question 15a: # Explanation: ## Step1: Check $ x = 0 $ in $ -2x < 10 $ Substitute $ x = 0 $: $ -2(0) = 0 < 10 $, true. ## Step2: Check $ x = 0 $ in $ -6 < -2x $ Substitute $ x = 0 $: $ -6 < -2(0) = 0 $, true. # Answer: Yes, $ x = 0 $ is a solution of both. Question 15b: # Explanation: ## Step1: Check $ x = 4 $ in $ -2x < 10 $ Substitute $ x = 4 $: $ -2(4) = -8 < 10 $, true. ## Step2: Check $ x = 4 $ in $ -6 < -2x $ Substitute $ x = 4 $: $ -6 < -2(4) = -8 $, false (since -6 > -8). # Answer: No, $ x = 4 $ is not a solution of both. Question 15c: # Explanation: ## Step1: Solve $ -2x < 10 $ Divide both sides by -2 (reverse inequality): $ x > -5 $. ## Step2: Solve $ -6 < -2x $ Divide both sides by -2 (reverse inequality): $ 3 > x $ or $ x < 3 $. ## Step3: Find $ x $ in $ -5 < x < 3 $ Let's pick $ x = 0 $ (already checked), $ x = 1 $: Check $ -2(1) = -2 < 10 $ (true), $ -6 < -2(1) = -2 $ (true). So $ x = 1 $ works. # Answer: $ x = 1 $ (any $ x $ between -5 and 3, not including -5 and 3, works) Question 16a: # Explanation: ## Step1: Let length be $ l $ and width be $ w $ Area of rectangle is $ A = lw $. Maximum area is 80, so $ lw \leq 80 $. # Answer: $ lw \leq 80 $ (where $ l $ is length, $ w $ is width) Question 16b: # Explanation: ## Step1: Choose $ l = 10 $, $ w = 8 $ Area: $ 10 \times 8 = 80 $ (which is ≤80). ## Step2: Choose $ l = 5 $, $ w = 10 $ Area: $ 5 \times 10 = 50 $ (≤80). ## Step3: Choose $ l = 20 $, $ w = 3 $ Area: $ 20 \times 3 = 60 $ (≤80). # Answer: - Dimensions: 10 ft (length) and 8 ft (width), Area: 80 sq ft - Dimensions: 5 ft (length) and 10 ft (width), Area: 50 sq ft - Dimensions: 20 ft (length) and 3 ft (width), Area: 60 sq ft

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Understand "at most"

Step2: Write the inequality

Step1: Translate "added to"

Step2: Translate "greater than"

Step1: Translate "multiplied by"

Step2: Translate "less than"

Answer: