Algebra
Calculus
Trigonometry
Matrix
Question
Solve the inequality
a≤−25+3
Alternative Form
a∈(−∞,−25+3]
Evaluate
−1−−(a×1)−a≥0
Find the domain
More Steps
Evaluate
−(a×1)≥0
Any expression multiplied by 1 remains the same
−a≥0
Change the signs on both sides of the inequality and flip the inequality sign
a≤0
−1−−(a×1)−a≥0,a≤0
Any expression multiplied by 1 remains the same
−1−−a−a≥0
Change the signs on both sides of the inequality and flip the inequality sign
−a+1+a≤0
Move the expression to the right side
−a≤−1−a
Separate the inequality into 2 possible cases
−a≤−1−a,−1−a≥0−a≤−1−a,−1−a<0
Solve the inequality
More Steps
Solve the inequality
−a≤−1−a
Square both sides of the inequality
−a≤(−1−a)2
Expand the expression
−a≤1+2a+a2
Move the expression to the left side
−a−(1+2a+a2)≤0
Subtract the terms
More Steps
Evaluate
−a−(1+2a+a2)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
−a−1−2a−a2
Subtract the terms
−3a−1−a2
−3a−1−a2≤0
Move the constant to the right side
−3a−a2≤0−(−1)
Add the terms
−3a−a2≤1
Evaluate
a2+3a≥−1
Add the same value to both sides
a2+3a+49≥−1+49
Evaluate
a2+3a+49≥45
Evaluate
(a+23)2≥45
Take the 2-th root on both sides of the inequality
(a+23)2≥45
Calculate
a+23≥25
Separate the inequality into 2 possible cases
a+23≥25a+23≤−25
Calculate
More Steps
Evaluate
a+23≥25
Move the constant to the right side
a≥25−23
Write all numerators above the common denominator
a≥25−3
a≥25−3a+23≤−25
Calculate
More Steps
Evaluate
a+23≤−25
Move the constant to the right side
a≤−25−23
Subtract the numbers
a≤−25+3
a≥25−3a≤−25+3
Find the union
a∈(−∞,−25+3]∪[25−3,+∞)
a∈(−∞,−25+3]∪[25−3,+∞),−1−a≥0−a≤−1−a,−1−a<0
Solve the inequality
More Steps
Evaluate
−1−a≥0
Move the constant to the right side
−a≥0+1
Removing 0 doesn't change the value,so remove it from the expression
−a≥1
Change the signs on both sides of the inequality and flip the inequality sign
a≤−1
a∈(−∞,−25+3]∪[25−3,+∞),a≤−1−a≤−1−a,−1−a<0
Since the left-hand side is always positive or 0,and the right-hand side is always negative,the statement is false for any value of a
a∈(−∞,−25+3]∪[25−3,+∞),a≤−1a∈∅,−1−a<0
Solve the inequality
More Steps
Evaluate
−1−a<0
Move the constant to the right side
−a<0+1
Removing 0 doesn't change the value,so remove it from the expression
−a<1
Change the signs on both sides of the inequality and flip the inequality sign
a>−1
a∈(−∞,−25+3]∪[25−3,+∞),a≤−1a∈∅,a>−1
Find the intersection
a≤−25+3a∈∅,a>−1
Find the intersection
a≤−25+3a∈∅
Find the union
a≤−25+3
Check if the solution is in the defined range
a≤−25+3,a≤0
Solution
a≤−25+3
Alternative Form
a∈(−∞,−25+3]
Show Solution
