How do you solve and write the following in interval notation: #-4

Question

How do you solve and write the following in interval notation: #-4<1/x<= 1
#?

Hazel Anglin · Accepted Answer

We get #(-infty,1/4)# or #[1,\infty)#

We are doing case work to solve this inequality:

a) #x>0# in this case we multiply the given inequality by #x# and we get

#-4x<1# and #1<=x#

so we have #x > -1/4# and #x>=1# so we get #x>=1# in this case.

b)#x<0# we multiply by #x<0#

so we get

#-4x>1>=x#

solving this we get

#x<-1/4# and #x<=1#

so we get the solution

#x<-1/4#

Abigail Allen · Answer

undefined To solve and write the inequality -4 < 1/x ≤ 1 in interval notation: 1. Find the critical points where the expression 1/x equals zero or is undefined. 2. Solve the inequalities individually for each interval created by the critical points. 3. Write the solution using interval notation. The critical point is x = 0, where 1/x is undefined. 1. For x > 0: Solve 1/x ≤ 1. - Multiply both sides by x (since x is positive): 1 ≤ x. - So, the solution for x > 0 is x ∈ [1, ∞). 2. For x < 0: Solve -4 < 1/x. - Multiply both sides by x (since x is negative, flip the inequality): -x < 4. - Divide both sides by -1 (flip the inequality): x > -4. - So, the solution for x < 0 is x ∈ (-∞, -4). Combine the solutions for both intervals: - The interval notation for the solution is: x ∈ (-∞, -4) ∪ [1, ∞).

How do you solve and write the following in interval notation: #-4&lt;1/x&lt;= 1 #?

How do you solve and write the following in interval notation: #-4<1/x<= 1 #?