Question

Express the set -4x+21 ≤ 3x +14 using interval notation

please give an explanation this is part of my review and i am having a hard time understanding what i’m doing wrong maybe i’m flipping the sign incorrectly or something in turn messing up my notation. thank you so much!

Answer 1

x ≥ 1

[1, ∞)

Explanation:

Note: 3 < 7 and 8 > 2 this is obvious

2,3,7,8 are on the RIGHT SIDE of the number zero "0"

now look at this sequence :

-3 > -7 and -8 < -2

the numbers are the same but thy are on the opposite side of the ZERO.

now notice that the <> signs flipped from the fist to the second set...

when you do these problems, "Multiplying or dividing by a negative number"

effectively moves you from one side of the zero to the other....

the "thing" you have to do is REMEBER AFTER EACH STEP...

IF YOU MULTIPLY OR DIVIDE BY A NEGATIVE NUMBER (not add or subtract)

YOU HAVE TO FLIP THE SIGN...

-4x+21 ≤ 3x +14

-4x ≤ 3x - 7 |||| subtracted 21 NO sign flip

-7x ≤ -7 ||| subtracted 3X NO sign flip

x ≥ 1 ||| Divided by "-1" SIGN MUST FLIP !!!

Answer 2

[1, ∞)

Explanation:

1) Solve for x

-4x + 21
`\leq` 3x + 14

21
`\leq` 7x + 14

7
`\leq` 7x

1
`\leq` x

2) 1 is the lowest possible value of x and since it's a smaller or equal to inequality, you use a square bracket. If it was 1 < x, it would be a round bracket. Since there isn't a boundary for the highest value of x, you write an infinity. When you have an infinity, it's always a round bracket.