Answer to Question #223005 in Linear Algebra for Vanessa

Question #223005


Solve the inequality √(x2-2x-8) ≤ -x+2


1
Expert's answer
2021-08-22T18:14:12-0400

x22x8x+2\sqrt{x^2-2x-8} ≤ -x+2



x22x8x+2\sqrt{x^2-2x-8} ≤ -x+2 ,x[,2][4,+],x∈[∞,− 2]∪[4,+∞]


Separate into possible cases


x22x8x+2,x+20\sqrt{x^2-2x-8} ≤ -x+2, -x+2≥0


x22x8x+2,x+2<0\sqrt{x^2-2x-8} ≤ -x+2, -x+2<0


Solve for inequality x


x6,x+20x≤6,-x+2≥0


x22x8x+2,x+2<0\sqrt{x^2-2x-8} ≤ -x+2, -x+2<0


x6,x2x≤6,x≤2


x22x8x+2,x+2<0\sqrt{x^2-2x-8} ≤ -x+2, -x+2<0


Since the left hand side is always positive or Zero and the right hand is always negative the statement is false for any value of x


x6,x2x≤6,x≤2

x,1+2<0x\in\varnothing,-1+2<0

x,1>2x\in\varnothing,1>2

Find the intersection

x[,2]x\in [-∞,-2]

xx\in\varnothing

Find the union

x[,2],x[,2][4,+]x∈[-∞,2],x∈[-∞,− 2]∪[4,+∞]


Find the intersection of the solution and the defined range


x[,2]x\in [-∞,-2]


Alternate form

x2x≤-2





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