Question

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

Answer 1

See a solution process below:

Explanation:

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]}

Answer 2

`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`)