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

2 Answers
Nov 7, 2017

See a solution process below:

Explanation:

Let's call "a number": xx

"The quotient" is the result of division.

In this problem, the numerator is: 3x3x

The denominator is: 44

So we can write:

(3x)/43x4

"is at least" means this is an inequality and specifically a >= inequality.

So, we can continue to write:

(3x)/4 >=3x4

And we can finish the inequality as:

(3x)/4 >= -163x416

To solve this, multiply each side of the inequality by color(red)(4)/color(blue)(3)43 to solve for nn while keeping the inequality balanced:

color(red)(4)/color(blue)(3) xx (3x)/4 >= color(red)(4)/color(blue)(3) xx -1643×3x443×16

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]}

Nov 7, 2017

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)

Explanation:

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)