4.1 Solve one variable equations and inequalities

Solve algebraically:√2x-1 -1x=-8

1) Add -1x to both sides : √2x-1 =x-8 2) Square both sides: 2x-1=(x-8)² 3) Factor the left side: 2x-1=x²-16x+64 4) Move everything from the right side to the left side by adding one and subtracting 2x: 0=x²-18x+65 5) Factor: 0=(x-13)(x-5) 6) Solve by adding 13 to 0=(x-13)and add 5 to 0=(x-5) Answers: x=13 and/or x=5

Solve algebraically: 2|3x+4|-10=12

1) Add 10 to both sides: 2|3x+4|=22 2) Divide 2 both sides: |3x+4|=11 3) Two solutions: 3x+4=11 and/or 3x+4=-11 4) Subtract 4 to both sides to the both equations: 3x=7 and/or 3x=-15 5) Divide 3 to both sides of the equations: x=7/3 and/or x=-5

Solve algebraically:√2x-1 -1x=0

1) Add 1x to both sides: √2x-1 =1x 2) Square both sides: 2x-1=1x 3) Subtract 2x to both sides: -1=-1x 4) Divide -1 to both sides: x=1

Solve algebraically: |2x-5|-6=15

1) Add 6 to both sides: |2x-5|=21 2) When it's an absolute value has two possible answers: 2x-5=21 and/or 2x-5=-21 3) Solve by adding 5 to both equations: 2x=26 and/or 2x=-16 4) Divide 2 to both equations: x=13 and/or x=-8

Solve algebraically: 5(√x-2 +1)=15

1) Divide 5 to both sides: √x-2 +1=3 2) Subtract 1 to both sides: √x-2=2 3) Square it both sides: x-2=4 4) Add 2 to both sides: x=6

Solve the system of equations: y=3x-5; y=-2x-15

1) Substitute one equation to make it equal to each other (answer for : 3x-5=-2x-15) 2) Add 2x to both sides: 5x-5=-15 3) Add 5 to both sides: 5x=-10 4) Divide 5 to both sides: x=-2 5) Now solve for y by substituting -2 to one of the given equations: y=3(-2)-5 6) Multiply 3(-2) first: y=-6-5 7) Subtract: y=-11 8) Guess and Check by substituting the answers to a given equation: -11=3(-2)-5 -11=-6-5 -11=-11 :)

Solve inequalities: 3x+2 ≥ x-6

1) Subtract 1x to both sides: 4x+2≥-6 2) Subtract 2 to both sides: 4x≥-8 3) Divide 4 to both sides: x≤-2

Solve inequalities: 1/3(3x-6)³ +4< 13

1) Subtract 4 to both sides:1/3(3x-6)³< 9 2) Divide 1/3 to both sides: (3x-6)³< 27 3)Cube it to both sides: 3x-6<3 4) Add 6 to both sides: 3x<9 5) Divide 3 both sides: Answer: x< 3

Solve inequalities: |x-4|<9

1) Two different equations: x-4<±9 2) Add 4 to both of the equations: x=13 and/or x=-5 (Answer:-5<x<13)

Solve inequalities :|1/2x-45|≥80

1) Two different equations: |1/2x-45|≥±80 2) Add 45 to both of the equations: 1/2x≥35 and/or 1/2x≥-35 3)Divide 2 to both sides for each equations:(Answer: x≥ 250 and/or x≤-70)

Solve algebraically: 2(x+3)²-5=-5

1)Add 5 to both sides: 2(x+3)²=0 2) Divide it by 2: (x+3)²=0 3) Write it as a factored form: 0=(x+3)(x+3) 4) Subtract 3 to the both equation: x=-3

Solve algebraically: 3(y+1)²-5=43

1)Add 5 to both sides: 3(y+1)²=48 2) Divide by 3: (y+1)²=16 3) Square root: y+1=±4 4) Subtract 1 to both equations: y=3 and/or y=-5

Solve algebraically: -3√2x-5 +7=-8

1)Subtract 7 to both sides: -3√2x-5 =-15 2) Divide -3 both sides: √2x-5= 5 3) Square root both sides: 2x-5=25 4) Add 5 to both sides: 2x=30 5) Divide 2: x=15