Algebra
Calculus
Trigonometry
Matrix
Question
Solve the inequality
x<3
Alternative Form
x∈(−∞,3)
Evaluate
x−3x2−∣x∣−12≥2x
Find the domain
More Steps
Evaluate
x−3=0
Move the constant to the right side
x=0+3
Removing 0 doesn't change the value,so remove it from the expression
x=3
x−3x2−∣x∣−12≥2x,x=3
Move the expression to the left side
x−3x2−∣x∣−12−2x≥0
Subtract the terms
More Steps
Evaluate
x−3x2−∣x∣−12−2x
Reduce fractions to a common denominator
x−3x2−∣x∣−12−x−32x(x−3)
Write all numerators above the common denominator
x−3x2−∣x∣−12−2x(x−3)
Multiply the terms
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Evaluate
2x(x−3)
Multiply the terms
(2x−6)x
Apply the distributive property
2x×x−6x
Multiply the terms
2x2−6x
x−3x2−∣x∣−12−(2x2−6x)
Subtract the terms
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Evaluate
x2−∣x∣−12−(2x2−6x)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
x2−∣x∣−12−2x2+6x
Subtract the terms
−x2−∣x∣−12+6x
x−3−x2−∣x∣−12+6x
x−3−x2−∣x∣−12+6x≥0
Change the signs on both sides of the inequality and flip the inequality sign
x−3x2+∣x∣+12−6x≤0
Separate the inequality into 2 possible cases
{x2+∣x∣+12−6x≥0x−3<0{x2+∣x∣+12−6x≤0x−3>0
Solve the inequality
More Steps
Evaluate
x2+∣x∣+12−6x≥0
Rewrite the expression
x2+∣x∣−6x≥−12
Separate the inequality into 2 possible cases
x2+x−6x≥−12,x≥0x2−x−6x≥−12,x<0
Evaluate
More Steps
Evaluate
x2+x−6x≥−12
Simplify the expression
x2−5x≥−12
Add the same value to both sides
x2−5x+425≥−12+425
Evaluate
x2−5x+425≥−423
Evaluate
(x−25)2≥−423
Calculate
x∈R
x∈R,x≥0x2−x−6x≥−12,x<0
Evaluate
More Steps
Evaluate
x2−x−6x≥−12
Simplify the expression
x2−7x≥−12
Add the same value to both sides
x2−7x+449≥−12+449
Evaluate
x2−7x+449≥41
Evaluate
(x−27)2≥41
Take the 2-th root on both sides of the inequality
(x−27)2≥41
Calculate
x−27≥21
Separate the inequality into 2 possible cases
x−27≥21x−27≤−21
Calculate
x≥4x−27≤−21
Calculate
x≥4x≤3
Find the union
x∈(−∞,3]∪[4,+∞)
x∈R,x≥0x∈(−∞,3]∪[4,+∞),x<0
Find the intersection
x≥0x∈(−∞,3]∪[4,+∞),x<0
Find the intersection
x≥0x<0
Find the union
x∈R
{x∈Rx−3<0{x2+∣x∣+12−6x≤0x−3>0
Solve the inequality
More Steps
Evaluate
x−3<0
Move the constant to the right side
x<0+3
Removing 0 doesn't change the value,so remove it from the expression
x<3
{x∈Rx<3{x2+∣x∣+12−6x≤0x−3>0
Solve the inequality
More Steps
Evaluate
x2+∣x∣+12−6x≤0
Rewrite the expression
x2+∣x∣−6x≤−12
Separate the inequality into 2 possible cases
x2+x−6x≤−12,x≥0x2−x−6x≤−12,x<0
Evaluate
More Steps
Evaluate
x2+x−6x≤−12
Simplify the expression
x2−5x≤−12
Add the same value to both sides
x2−5x+425≤−12+425
Evaluate
x2−5x+425≤−423
Evaluate
(x−25)2≤−423
Calculate
x∈/R
x∈/R,x≥0x2−x−6x≤−12,x<0
Evaluate
More Steps
Evaluate
x2−x−6x≤−12
Simplify the expression
x2−7x≤−12
Add the same value to both sides
x2−7x+449≤−12+449
Evaluate
x2−7x+449≤41
Evaluate
(x−27)2≤41
Take the 2-th root on both sides of the inequality
(x−27)2≤41
Calculate
x−27≤21
Separate the inequality into 2 possible cases
{x−27≤21x−27≥−21
Calculate
{x≤4x−27≥−21
Calculate
{x≤4x≥3
Find the intersection
3≤x≤4
x∈/R,x≥03≤x≤4,x<0
Find the intersection
x∈/R3≤x≤4,x<0
Find the intersection
x∈/Rx∈∅
Find the union
x∈/R
{x∈Rx<3{x∈/Rx−3>0
Solve the inequality
More Steps
Evaluate
x−3>0
Move the constant to the right side
x>0+3
Removing 0 doesn't change the value,so remove it from the expression
x>3
{x∈Rx<3{x∈/Rx>3
Find the intersection
x<3{x∈/Rx>3
Find the intersection
x<3x∈/R
Find the union
x<3
Check if the solution is in the defined range
x<3,x=3
Solution
x<3
Alternative Form
x∈(−∞,3)
Show Solution
