Mathematics, 13.03.2022 23:00 gracieisweird12

Hello Guys, I really need help with these questions. Here they are. Solve each absolute value inequality and show its solution set.
(a) |x|<7
(b) |x+3|<9
(c) |y-8|>11
(g) |3x-1| greater or equal to 18
|4y+3| less than or equal to 13
Please help me with these. WIll give you 20 pts. I will award for the best answer! Make the answers in numbers (inequalities)

Answers: 2

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Answers

Answer from: Bunnybear8099

See below.

Step-by-step explanation:

It can really help to think when you see |expression| that it means the distance from expression to zero.

(a) |x| < 7 means the distance from x to 0 is less than 7. That puts x between -7 and 7. The solution set is -7 < x < 7.

(b) |x + 3| < 9 means that the distance from x + 3 to 0 is less than 9. That puts x + 3 between -9 and 9:

-9 < x + 3 < 9 Now subtract 3 from all three parts.

-12 < x < 6

(c) |y - 8| > 11 means that the distance from y - 8 to 0 is more than 11 units. That puts y - 8 in one of two places: left of -11 or right of 11.

$y-8 < -11 \text{ or } y-8 11\\\\y < -3 \text{ or } y 19$

(g) $|3x-1| \ge 18$ means that the distance from 3x - 1 to 0 is more than (or equal to) 18. Another way to say it is, 3x - 1 is farther from 0 than 18 units. That puts 3x - 1 in one of two places: to the left of -18 or to the right of 18.

$3x-1 \le -18 \text{ or } 3x-1 \ge 18\\\\3x \le -17 \text{ or } 3x \ge 19\\\\x \le -\frac{17}{3} \text{ or } x \ge \frac{19}{3}$

$|4y+3| \le 13$ means that 4y + 3 is closer to 0 than 13; it is between -13 and 13.

$-13 \le 4y+3 \le 13\\\\-16 \le 4y \le 10\\\\-4 \le y \le \frac{5}{2}$

(That last fraction is 10/4 simplified.)

Answer from: lillianbrowning10

a. -7 < x < 7
b. -12 < x < 6
c. Two answers are: y < -3 or y > 19
d. Two answers are x ≤ -17/3 or x ≥ 19/3
e. -4 ≤ y ≤ 5/2

Step-by-step explanation:

Given the following questions:

Question one:

|x| < 7
x is less than 7
=-7 < x < 7

Question two:

|x+3| < 9
x+3 -9
3-3=0
-9+-3=-12
x -12

x+3 < 9
3-3=0
9-3=6
x < 6
x is greater than -12, while less than 6
-12 < x < 6

Question three:

|y-8| 11
y-8 < -11
-8+8=0
-11+8=-3
y < -3

y-8 11
-8+8=0
11+8=19
y 19
Two answers are: y < -3 or y > 19

Question four:

|3x - 1| ≥ 18
3x - 1 ≤ -18
-1 + 1 = 0
-18 + 1 = -17
3x ≤ -17
x ≤ -17/3

3x - 1 ≥ 18
-1 + 1 = 0
18 + 1 = 19
3x ≥ 19
x ≥ 19/3
Two answers are: x ≤ -17/3 or x ≥ 19/3

Question five:

|4y + 3| ≤ 13
|4y + 3| ≤ 13 = -13 ≤ 4y + 3 ≤ 13
-13 ≤ 4y + 3 ≤ 13
4y + 3 ≥ -13
3 - 3 = 0
-13 + - 3 = -16
4y ≥ -16
4 ÷ - 16 = -4
y ≥ -4

4y + 3 ≤ 13
3 + -3 = 0
13 - 3 = 10
4y ≤ 10 = 10/4 ÷ 2 = 5/2
y ≤ 5/2
-4 ≤ y ≤ 5/2

Hope this helps.