Algebra
Calculus
Trigonometry
Matrix
Question
Solve the inequality
x≥32
Alternative Form
x∈[32,+∞)
Evaluate
log10(2x×1)≤log10(x4)
Find the domain
More Steps
Evaluate
{2x×1>0x4>0
Calculate
More Steps
Evaluate
2x×1>0
Multiply the terms
2x>0
Rewrite the expression
x>0
{x>0x4>0
Calculate
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Evaluate
x4>0
Since the left-hand side is always positive or 0,and the right-hand side is always 0,the statement is true for any value of x,except when x4=0
x4=0
The only way a power can be 0 is when the base equals 0
x=0
Exclude the impossible values of x
x=0
{x>0x=0
Find the intersection
x>0
log10(2x×1)≤log10(x4),x>0
Multiply the terms
log10(2x)≤log10(x4)
For 10>1 the expression log10(2x)≤log10(x4) is equivalent to 2x≤x4
2x≤x4
Move the expression to the left side
2x−x4≤0
Factor the expression
x(2−x3)≤0
Separate the inequality into 2 possible cases
{x≥02−x3≤0{x≤02−x3≥0
Solve the inequality
More Steps
Evaluate
2−x3≤0
Rewrite the expression
−x3≤−2
Change the signs on both sides of the inequality and flip the inequality sign
x3≥2
Take the 3-th root on both sides of the equation
3x3≥32
Calculate
x≥32
{x≥0x≥32{x≤02−x3≥0
Solve the inequality
More Steps
Evaluate
2−x3≥0
Rewrite the expression
−x3≥−2
Change the signs on both sides of the inequality and flip the inequality sign
x3≤2
Take the 3-th root on both sides of the equation
3x3≤32
Calculate
x≤32
{x≥0x≥32{x≤0x≤32
Find the intersection
x≥32{x≤0x≤32
Find the intersection
x≥32x≤0
Find the union
x∈(−∞,0]∪[32,+∞)
Check if the solution is in the defined range
x∈(−∞,0]∪[32,+∞),x>0
Solution
x≥32
Alternative Form
x∈[32,+∞)
Show Solution
