How do you solve #|5x+2|=|3x-4|#?

Question

How do you solve #|5x+2|>=|3x-4|#?

Kennedy Adams · Accepted Answer

Solution : # x<= -3 or x>=1/4# OR # (-oo,-3]uu[1/4,oo)# #|5x+2| >= |3x-4|# squaring both sides we get # (5x+2)^2 >= (3x-4)^2 # or #25x^2+20x+4 >= 9x^2 -24x +16# or #25x^2+20x+4 - 9x^2 +24x -16 >=0# or #16x^2+44x-12>=0# or #4x^2+11x-3>=0# or #(4x-1)(x+3).>=0#. Critical points are #x=-3 and x=1/4# Sign Change: For #x=-3 or x=1/4; (4x-1)(x+3)=0# For #x<-3 ; (4x-1)(x+3)= (-*-)= (+) ; (4x-1)(x+3)>0 # For #-3 < x < 1/4 ; (4x-1)(x+3)= (-*+)= (-) ; (4x-1)(x+3)<0 # For #x>1/4 ; (4x-1)(x+3)= (+*+)= (+) ; (4x-1)(x+3)>0 # So solution : # x<= -3 or x>=1/4# or # (-oo,-3]uu[1/4,oo)# The graph also confirms the findings. graph{4x^2+11x-3 [-40, 40, -20, 20]}

Eva Adamson · Answer

undefined To solve the inequality |5x + 2| ≥ |3x - 4|, you need to consider two cases:

1. When 5x + 2 ≥ 0 and 3x - 4 ≥ 0:
   - Solve 5x + 2 ≥ 3x - 4: 
     5x + 2 ≥ 3x - 4
     2x ≥ -6
     x ≥ -3

2. When 5x + 2 < 0 and 3x - 4 < 0:
   - Solve 5x + 2 ≤ -(3x - 4): 
     5x + 2 ≤ -3x + 4
     8x ≤ 2
     x ≤ 1/4

Therefore, the solution set for the inequality |5x + 2| ≥ |3x - 4| is x ≤ 1/4 or x ≥ -3.

Jace Anderson · Answer

undefined To solve the inequality $|5x+2| \geq |3x-4|$, we consider different cases based on the possible signs of $5x+2$ and $3x-4$.

Case 1: $5x+2$ and $3x-4$ are both non-negative or both non-positive:
In this case, the absolute value inequality simplifies to $5x+2 \geq 3x-4$. Solve for $x$ to find the solution.

Case 2: $5x+2$ is non-negative and $3x-4$ is non-positive:
In this case, the absolute value inequality simplifies to $5x+2 \geq -(3x-4)$. Solve for $x$ to find the solution.

Case 3: $5x+2$ is non-positive and $3x-4$ is non-negative:
In this case, the absolute value inequality simplifies to $-(5x+2) \geq 3x-4$. Solve for $x$ to find the solution.

Case 4: $5x+2$ and $3x-4$ are both non-zero with opposite signs:
In this case, the absolute value inequality simplifies to $-(5x+2) \geq -(3x-4)$. Solve for $x$ to find the solution.

Once you have solved each case, you should consider the intersection of all solutions to obtain the final solution set for the inequality.

How do you solve #|5x+2|&gt;=|3x-4|#?

How do you solve #|5x+2|>=|3x-4|#?