How do you solve an inequality with 3 terms?

Question

I apologize; but I have forgotten over the years how to solve inequalities with 3 terms.

When I use the < symbol followed directly by the - symbol, it will appear like <-. Please take that as "Less Than or Equal To". Please take -> as "Greater Than or Equal To".

The question I am attempting to solve is:

-9 <- -3x+15 <- 12

Negative Nine is Less than or Equal to Three X plus Fifteen which is Less than or Equal to 12.

Any assistance is gladly appreciated!

Thanks!

Answer 1

(Note: Hold down the 'alt' key and type 243 to get ≤. Use 242 to get ≥)

-9 ≤ -3x+15 ≤ 12 Subtract 15 from all three terms
-24 ≤ -3x ≤ -3 Divide all three terms by -3 (and remember to reverse the inequalities)
8 ≥ x ≥ 1 Or, if you prefer
1 ≤ x ≤ 8

Treat them like any other equation except that when you multiply or divide by a negative quantity, it reverses the direction of the inequality.

Hope that helps.


Doug

PSSSSTTT!!!!!!!!!!! DR -24/-3 is *NOT* 6, it's 8. But it's good to know I'm not the only one who does things like that

Answer 2

I'm afraid most of the answers thus far, while correct in breaking the inequality into two inequalities and solving them similar to equations (with equal signs), have fallen to the common misunderstanding concerning multiplying and dividing the equations by negative signs/numbers; there exists a very significant difference when working with inequalities. The inequality sign switches directions with the operation of multiplying by a negative sign/number.

For instance: -3*x > -3 becomes -3*x / -3 < -3 / -3

Notice the inequality sign switched from pointing to the right to pointing to the left. The same applies to >= switching to <=, which read "greater than or equal" and "less than or equal," respectively. Also, note that >= is the standard notation and much less confusing than -> (=> is better, but optimal because it reads "equal to or greater than", instead of the traditionally spoken "greater than or equal to").

The solution of the previous examples concludes:
x < 1

If the original inequality had read -9 > -3*x > -3 then it could be divided into -9 > -3*x and -3*x > -3 (the latter was just solved above) or solved as one piece with all three parts being treated equally, like this:

-9 > -3*x > -3
-9 / -3 < -3*x / -3 < -3 / -3
Notice that all three parts are divided by -3 and that all three inequality signs are switched because of the division (it applies equally to multiplication) of a negative number.
3 < x < 1 The division is resolved from the previous inequality in this step.
Therefore x > 3 and x < 1
Notice that the inequality sign in the first part of the answer points in a different direction because the x is placed before the 3. The inequality sign still points at the same number, 3.
Thus, 3 < x is the same as x > 3. In words, 3 is less than x is the same as x is greater than 3. If this doesn't make sense after some consideration, you may just as well leave your answer as 3 < x < 1 or in the less common form of 3 < x and x < 1

The equals sign in the equality continues through unaffected by mult/div of negative signs/numbers. Thus a modification of our first example reveals:
For instance: -3*x >= -3 becomes -3*x / -3 <= -3 / -3, which yields x <= 1.

Answer 3

Your problem is to find what values of x will make the equation -3x+15 equal to values between -9 and 12 (inclusive)?

First solve for the lower limit:
-3x+15 = -9
-3x = -24
x = -24/-3 = 6

Next solve for the upper limit:
-3x + 15 = 12
-3x = -3
x = -3/-3 = 1

Thus x can be any value between 1 and 6.

Answer 4

-9 <- -3x+15 <- 12
First you subtract 15 from all three like so:
-9-15 <- -3x+15-15 <- 12-15
Simplify:
-24<- -3x <- -3
Divide all three sides by three, remembering that in an inequality, division or multiplication by a negative number flips the sign in the opposite direction. Then simplify.
So, it becomes:
-8 >- x >- 1
I'm quite positive that this is the answer. Hope I was clear.

Answer 5

-9 ≤ 3x + 15 ≤ 12

- 9 - 15 ≤ 3x + 15 - 15 ≤ 12 - 15

subtracting - 15 from all three terms

- 24 ≤3x ≤ - 3

- 24/3 ≤ 3x/3 ≤ -3/3

Dividing 3 from all three terms

- 8 ≥ x ≥ -1

Once the division is completed the inequalitie sign changes.

The answer is - 8 ≥ x ≥ -1

Answer 6

I will use <= to denote "less than or equal to" and >= to denote "greater than or equal to.

Your inequality can be broken down into two seperate inequalities

-9 <= -3x + 15

and

-3x + 15 <= 12

The first inequality simplifies to

8 >= x

The second inequality simplifies to

x >= 1

So the full solution is 1 <= x <= 8

Answer 7

-9< 3x+15 < 12
is the same as
-9< 3x+15 AND 3x+15<12...

do the rest yourself

You know how to solve both inequations independantely, don't you? then you can solve them both independantly and merge the answer properly.

When in front of a problem, transform it in something simpler to solve.


"The same equations have the same solutions." (Thus when you have solved a mathematical problem, you can re-use the solution in another physical situation. Feynman was skilled in transforming a problem into one that he could solve.)
(taken from the wikipedia article on richard feynman)

Answer 8

19h-7=23+14h ==> Subtract 14 h from the two aspects. 5h-7=23 ==> upload 7 to the two aspects. 5h=30 ==> Divide the two aspects by utilising 5 h=6 you double examine. -5 subtract 3 -8-8 subract, upload, divide, and multiply the two aspects till you get basically the variable. hint: question #5 is that q could be any quantity question #6 is have not have been given any answer

Answer 9

When solving for 3 variables, I use a matrix on a graphing calculator, solve the inequalities and enter them in, it will give you the answers. I'm not sure how to do it by hand, not sure that anyone does anymore. Oh the wonders of the calculator!

Answer 10

-9< 3x+15
-9-15<3x
-24<3x
-8
-3x+15<12
-3x<-15+12
-3x<-3
x<1

Ans: -8 pablo forgot his minus sign