Solving Linear Inequality

[holmes1172]

To give the solution set in interval notation for -3x-8≤ 7 can I take the eight and write this as -8-3x-8≤
7 so I can subtract it as a first step?

 

[Quaid]

To give the solution set in interval notation for

-3x - 8 ≤ 7

can I take the eight and write this as

-8 - 3x - 8 ≤ 7

so I can subtract it as a first step?

No, you may not add -8 to the lefthand side without also adding -8 to the righthand side.

If you change the value of only one side of an equation or inequality, you've changed the exercise.


The first step could be to isolate the -3x term, by adding 8 to each side.

Then you can solve for x, by dividing each side by -3.

REMEMBER! When dividing or multiplying both sides of an inequality by a negative value, you must change the direction of the inequality symbol.


Please give it another go. :cool:


TIP: Note how I altered your typing, in the quote above. It is much easier for everyone to read, when you (1) put spaces around operator symbols and (2) type each math statement on its own line. Thank you.

 

[holmes1172]

No, you may not add -8 to the lefthand side without also adding -8 to the righthand side.

If you change the value of only one side of an equation or inequality, you've changed the exercise.


The first step could be to isolate the -3x term, by adding 8 to each side.

Then you can solve for x, by dividing each side by -3.

REMEMBER! When dividing or multiplying both sides of an inequality by a negative value, you must change the direction of the inequality symbol.


Please give it another go. :cool:


TIP: Note how I altered your typing, in the quote above. It is much easier for everyone to read, when you (1) put spaces around operator symbols and (2) type each math statement on its own line. Thank you.

Sure thing, and thank you. I appreciate your help.
So, let's try this again:
-3x -8 ≤ 7

-3x + 8 -8 ≤ 7 -8 + 8

-3x ≤ 7

-3 / -3 ≤ 7 / -3 (I'm not sure I did something right here. That 7 is not divisible by the -3 and has me thinking I did something wrong...

 

[Nazariy]

Sure thing, and thank you. I appreciate your help.
So, let's try this again:
-3x -8 ≤ 7

-3x + 8 -8 ≤ 7 -8 + 8

-3x ≤ 7

-3 / -3 ≤ 7 / -3 (I'm not sure I did something right here. That 7 is not divisible by the -3 and has me thinking I did something wrong...

If you add eight to one side, you add it to the other. So you do not have -8 on the right side there.

You have:

-3x + 8 -8 ≤ 7 + 8

 

[holmes1172]

If you add eight to one side, you add it to the other. So you do not have -8 on the right side there.

You have:

-3x + 8 -8 ≤ 7 + 8

Well, I'm feeling fairly bright right now. :roll: I have no idea why I put that -8 there. Sorry. So then we would be at

-3x / -3 ≤ 15/ -3

x = -5?

 

[Nazariy]

Well, I'm feeling fairly bright right now. :roll: I have no idea why I put that -8 there. Sorry. So then we would be at

-3x / -3 ≤ 15/ -3

x = -5?

Nope.

When we divide by a negative sign, the inequality changes its sign.

i.e. >


You can check whether the initial inequality holds if you plug in numbers that are greater or equal to negative five, if it holds then you have probably solved it correctly.

For example if you plug -5, you get > -3*(-5)-8≤ 7 which is 7 ≤ 7. Thus correct so far. If you plug -4 or more what happens? Let's say we plug in 0, then -3*0-8≤ 7, correct, since -8 ≤ 7. Conversely, you can see that if you plug numbers that are less than negative 5, i.e. more negative, then the inequality does not hold, for example, suppose x=-10: -3*(-10)-8≤ 7 > 30-8≤7, 22 is NOT less or equal to 7!

 

[holmes1172]

Nope.

When we divide by a negative sign, the inequality changes its sign.

i.e. >

View attachment 3730
You can check whether the initial inequality holds if you plug in numbers that are greater or equal to negative five, if it holds then you have probably solved it correctly.

For example if you plug -5, you get > -3*(-5)-8≤ 7 which is 7 ≤ 7. Thus correct so far. If you plug -4 or more what happens? Let's say we plug in 0, then -3*0-8≤ 7, correct, since -8 ≤ 7. Conversely, you can see that if you plug numbers that are less than negative 5, i.e. more negative, then the inequality does not hold, for example, suppose x=-10: -3*(-10)-8≤ 7 > 30-8≤7, 22 is NOT less or equal to 7!

Ah, I see. Do we only change the inequality sign when we divide or is there another time like when we subtract?

 

[Nazariy]

Ah, I see. Do we only change the inequality sign when we divide or is there another time like when we subtract?

We change the sign if we multiply or divide with negative sign.

 

[holmes1172]

We change the sign if we multiply or divide with negative sign.

Got it. I'll remember that. This actually gets kind of fun once you start understanding.

 

[JeffM]

Ah, I see. Do we only change the inequality sign when we divide or is there another time like when we subtract?

The direction of inequality changes when you multiply or divide both sides of the inequality by a negative number. The direction of inequality is not affected by adding or subtracting any number to both sides of the inequality nor by multiplying or dividing both sides of the inequality by a non-negative number.

Let's see some examples

\(\displaystyle 5 < 10 \implies 5 + 3 < 10 + 3 \implies 8 < 13.\) You buy that? No change in the direction of inequality.

\(\displaystyle 5 < 10 \implies 5 - 3 < 10 - 3 \implies 2 < 7.\) You buy that? No change in the direction of inequality.

\(\displaystyle 5 < 10 \implies 2 * 5 < 2 * 10\implies 10 < 20.\) You buy that? No change in the direction of inequality.

\(\displaystyle 5 < 10 \implies 5 \div 5 < 10 \div 5 \implies 1 < 2.\) You buy that? No change in the direction of inequality.

\(\displaystyle 5 * (-2) = - 10,\ but\ 10 * (-2) = - 20\ and\ - 10 > - 20.\) It is warmer when it is 10 below zero than when it is 20 below.

\(\displaystyle So\ 5 < 10 \implies (-2) * 5 > (-2) * 10 \implies - 10 > - 20.\) Change in the direction of inequality.

\(\displaystyle 5 \div (-5) = - 1,\ but\ 10 \div (-5) = - 2\ and\ - 1 > - 2.\) It is warmer when it is 1 below zero than when it is 2 below.

\(\displaystyle So\ 5 < 10 \implies 5 \div (-5) > 10 \div (- 5) \implies - 1 > - 2.\) Change in the direction of inequality.