Solve |2x-1|>4 as an interval

Answer 1

Hello Victoria

This question is based using your understanding of the usage of 'absolute values' and their effect on intervals. When solving for what is inside an absolute value it is good to know how the bars function...

Whatever is inside of the absolute value sign | part of equation in this case 2x-1 | the part of your equation will become positive or 0 no matter what happens on the inside of the absolute value. This is due to absolute value acting finding the absolute or total distance from 0 on the number line.

Therefore, we can break the inequality into two separate equations to find the interval where x is valid.

Let's say the first equation is initially positive and the absolute value changes nothing...

2x-1 > 4

Solve this algebraically by adding the 1 to both sides...

2x -1 +1 > 4 + 1

2x>5

Next solve for x by dividing both sides by 2 in order to separate x

2x/2 > 5/2

x>2.5

Check if this is a solution by plugging x into the initial equation.

|2*2.5 - 1|>4

|5-1|>4

4>4 # note 4 = 4 however this is not the answer on you interval since you want what is greater than 4 not including 2.5 so your inequality graph can have an open circle at 2.5 with a line moving towards infinity or the right hand side of the inequality graph.

Now, lets solve for the opposite version of the inequality since we know that the absolute value can change negative numbers to positive numbers we will be looking for numbers that are lesser than -4.

Here is our equation...

2x-1 <-4 #consider this on the inside of the absolute value everything stayed the same however the outside completely flipped from greater than to less than and positive to negative.

Again, we will cycle through our algebra tools to separate x...

2x-1+1 <-4+1

2x<-3

2x/2<-3/2

x<-1.5

Again, be careful with the answer since we do not have the line underneath the lesser than sign we will be not including -1.5 in the answer.

This will be interval notation using the U (union sign) to show that there are multiple sections to the answer in this case only 2 sections.

( signifies non-inclusion or -∞ on the left hand side

) signifies non-inclusion or ∞ on the right hand side

[ signifies inclusion on the left hand side

] signifies inclusion on the right hand side

the final answer is (-∞, -1.5)U(2.5, ∞)

Answer 2

Since absolute value is the distance from zero, eight and negative eight have the same absolute value, since they are both eight from zero.

Four and negative four both have an absolute value of four, but to be farther than four, the inside has to be greater than four orrrr less than negative four (the absolute value of negative three is less than four.

2x - 1 > 4

2x > 5

x > 5/2

2x - 1 < -4

2x < -3

x < -3/2

(-∞,-3/2) ∪ (5/2,∞)

Answer 3

|2x-1|>4

2x>5

2x>-5

x>2.5

x>-2.5

Those lines stand for "absolute value"; or distance from "0".