How do you solve "quotient of three times a number and 4 is at least -16" and graph the solution on a number line?

Question

Hazel Anglin · Accepted Answer

See a solution process below:

Let's call "a number": #x# "The quotient" is the result of division. In this problem, the numerator is: #3x# The denominator is: #4# So we can write: #(3x)/4# "is at least" means this is an inequality and specifically a #>=# inequality. So, we can continue to write: #(3x)/4 >=# And we can finish the inequality as: #(3x)/4 >= -16# To solve this, multiply each side of the inequality by #color(red)(4)/color(blue)(3)# to solve for #n# while keeping the inequality balanced: #color(red)(4)/color(blue)(3) xx (3x)/4 >= color(red)(4)/color(blue)(3) xx -16# #cancel(color(red)(4))/cancel(color(blue)(3)) xx (color(blue)(cancel(color(black)(3)))x)/color(red)(cancel(color(black)(4))) >= -64/3# #x >= -64/3# To graph this we will draw a vertical line at #-64/3# on the horizontal axis. The line will be a solid line because the inequality operator contains an "or equal to" clause. We will shade to the right side of the line because the inequality operator also contains a "greater than" clause: graph{x>=-64/3 [-30, 30, -15, 15]}

Hazel Anglin · Answer

#x>=-21 1/3#
To graph this on a number line, you would make a solid dot on the point #(-21 1/3)#, with the line moving to the right (#rarr#)

First, let's analyze what each word means. "quotient (#-:#) of three times a number (#3x#) and four (#+4#) is at least -16 (#>=-16#)" Now take out the numbers. #3x-:4>=-16# Now to find the possibilities of #x#, balance the inequality. #3x-:4>=-16# Multiply both sides by 4. #3x>=-64# Divide both sides by 3. #x>=-21 1/3# To graph this on a number line, you would make a solid dot on the point #(-21 1/3)#, with the line moving to the right (#rarr#)

Eva Abramson · Answer

undefined To solve the inequality "quotient of three times a number and 4 is at least -16," first, translate the sentence into an algebraic expression. Then, solve the inequality algebraically to find the solution set. Finally, represent the solution set on a number line graphically.