Algebra
Calculus
Trigonometry
Matrix
Question
∣x−2∣<∣x−4∣
Solve the inequality
x<3
Alternative Form
x∈(−∞,3)
Evaluate
∣x−2∣<∣x−4∣
Rearrange the terms
∣x−2∣−∣x−4∣<0
Separate the inequality into 4 possible cases
x−2−(x−4)<0,x−2≥0,x−4≥0x−2−(−(x−4))<0,x−2≥0,x−4<0−(x−2)−(x−4)<0,x−2<0,x−4≥0−(x−2)−(−(x−4))<0,x−2<0,x−4<0
The statement is false for any value of x
More Steps
Evaluate
x−2−(x−4)<0
Remove the parentheses
x−2−x+4<0
Simplify the expression
2<0
The statement is false for any value of x
x∈∅
x∈∅,x−2≥0,x−4≥0x−2−(−(x−4))<0,x−2≥0,x−4<0−(x−2)−(x−4)<0,x−2<0,x−4≥0−(x−2)−(−(x−4))<0,x−2<0,x−4<0
Evaluate
More Steps
Evaluate
x−2≥0
Move the constant to the right side
x≥0+2
Removing 0 doesn't change the value,so remove it from the expression
x≥2
x∈∅,x≥2,x−4≥0x−2−(−(x−4))<0,x−2≥0,x−4<0−(x−2)−(x−4)<0,x−2<0,x−4≥0−(x−2)−(−(x−4))<0,x−2<0,x−4<0
Evaluate
More Steps
Evaluate
x−4≥0
Move the constant to the right side
x≥0+4
Removing 0 doesn't change the value,so remove it from the expression
x≥4
x∈∅,x≥2,x≥4x−2−(−(x−4))<0,x−2≥0,x−4<0−(x−2)−(x−4)<0,x−2<0,x−4≥0−(x−2)−(−(x−4))<0,x−2<0,x−4<0
Evaluate
More Steps
Evaluate
x−2−(−(x−4))<0
Remove the parentheses
x−2+x−4<0
Simplify the expression
2x−6<0
Move the constant to the right side
2x<0+6
Removing 0 doesn't change the value,so remove it from the expression
2x<6
Divide both sides
22x<26
Divide the numbers
x<26
Divide the numbers
More Steps
Evaluate
26
Reduce the numbers
13
Calculate
3
x<3
x∈∅,x≥2,x≥4x<3,x−2≥0,x−4<0−(x−2)−(x−4)<0,x−2<0,x−4≥0−(x−2)−(−(x−4))<0,x−2<0,x−4<0
Evaluate
More Steps
Evaluate
x−2≥0
Move the constant to the right side
x≥0+2
Removing 0 doesn't change the value,so remove it from the expression
x≥2
x∈∅,x≥2,x≥4x<3,x≥2,x−4<0−(x−2)−(x−4)<0,x−2<0,x−4≥0−(x−2)−(−(x−4))<0,x−2<0,x−4<0
Evaluate
More Steps
Evaluate
x−4<0
Move the constant to the right side
x<0+4
Removing 0 doesn't change the value,so remove it from the expression
x<4
x∈∅,x≥2,x≥4x<3,x≥2,x<4−(x−2)−(x−4)<0,x−2<0,x−4≥0−(x−2)−(−(x−4))<0,x−2<0,x−4<0
Evaluate
More Steps
Evaluate
−(x−2)−(x−4)<0
Remove the parentheses
−x+2−x+4<0
Simplify the expression
−2x+6<0
Move the constant to the right side
−2x<0−6
Removing 0 doesn't change the value,so remove it from the expression
−2x<−6
Change the signs on both sides of the inequality and flip the inequality sign
2x>6
Divide both sides
22x>26
Divide the numbers
x>26
Divide the numbers
More Steps
Evaluate
26
Reduce the numbers
13
Calculate
3
x>3
x∈∅,x≥2,x≥4x<3,x≥2,x<4x>3,x−2<0,x−4≥0−(x−2)−(−(x−4))<0,x−2<0,x−4<0
Evaluate
More Steps
Evaluate
x−2<0
Move the constant to the right side
x<0+2
Removing 0 doesn't change the value,so remove it from the expression
x<2
x∈∅,x≥2,x≥4x<3,x≥2,x<4x>3,x<2,x−4≥0−(x−2)−(−(x−4))<0,x−2<0,x−4<0
Evaluate
More Steps
Evaluate
x−4≥0
Move the constant to the right side
x≥0+4
Removing 0 doesn't change the value,so remove it from the expression
x≥4
x∈∅,x≥2,x≥4x<3,x≥2,x<4x>3,x<2,x≥4−(x−2)−(−(x−4))<0,x−2<0,x−4<0
The statement is true for any value of x
More Steps
Evaluate
−(x−2)−(−(x−4))<0
Remove the parentheses
−x+2+x−4<0
Simplify the expression
−2<0
The statement is true for any value of x
x∈R
x∈∅,x≥2,x≥4x<3,x≥2,x<4x>3,x<2,x≥4x∈R,x−2<0,x−4<0
Evaluate
More Steps
Evaluate
x−2<0
Move the constant to the right side
x<0+2
Removing 0 doesn't change the value,so remove it from the expression
x<2
x∈∅,x≥2,x≥4x<3,x≥2,x<4x>3,x<2,x≥4x∈R,x<2,x−4<0
Evaluate
More Steps
Evaluate
x−4<0
Move the constant to the right side
x<0+4
Removing 0 doesn't change the value,so remove it from the expression
x<4
x∈∅,x≥2,x≥4x<3,x≥2,x<4x>3,x<2,x≥4x∈R,x<2,x<4
Find the intersection
x∈∅x<3,x≥2,x<4x>3,x<2,x≥4x∈R,x<2,x<4
Find the intersection
x∈∅2≤x<3x>3,x<2,x≥4x∈R,x<2,x<4
Find the intersection
x∈∅2≤x<3x∈∅x∈R,x<2,x<4
Find the intersection
x∈∅2≤x<3x∈∅x<2
Solution
x<3
Alternative Form
x∈(−∞,3)
Show Solution
