Question

Answer 1

`a, x^2 + 2x - 3 - x^2 + 9 >0`

`<=> 2x + 6 >0`

`<=> 2x > -6`

`<=> x > -3`.

`b, (x+2)/3 - (3x-1)/5 <-2`

`<=> (5x+10-9x+3)/15<-2`

`<=> -4x + 13 <-30`

`<=> -4x <-43`.

`<=> 4x > 43.`

`<=> x >43/4`.

`c, x^2-9 - x^2 - 2x <=0`

`<=> 9 + 2x >=0`

`<=> 2x >=-9`

`<=> x >=-9/2`.

`d, (x-1)/4 -1 > (x+1)/3 + 8`

`<=> (3x-3 -4x-4)/12>9`

`<=> -x - 7 >108`.

`<=> -x > 115`.

`<=> x <-115.`

Answer 2

a) \(\left(x-1\right)\left(x+3\right)-x^2+9>0\)

\(\Leftrightarrow\left(x-1\right)\left(x+3\right)-\left(x+3\right)\left(x-3\right)>0\)

\(\Leftrightarrow\left(x+3\right)\left[\left(x-1\right)-\left(x-3\right)\right]>0\)

\(\Leftrightarrow2\left(x+3\right)>0\)

\(\Leftrightarrow x+3>0\)

\(\Leftrightarrow x>-3\)

b) \(\dfrac{x+2}{3}-\dfrac{3x-1}{5}< -2\)

\(\Leftrightarrow\dfrac{5\left(x+2\right)}{15}-\dfrac{3\left(3x-1\right)}{15}< \dfrac{-2\cdot15}{15}\)

\(\Leftrightarrow5x+10-9x+3< -30\)

\(\Leftrightarrow-4x+13< -30\)

\(\Leftrightarrow-4x< -43\)

\(\Leftrightarrow x>\dfrac{43}{4}\)

Answer 3

\(a,\left(x-1\right)\left(x+3\right)-x^2+9>0\\ \Leftrightarrow x^2+2x-3-x^2+9>0\\ \Leftrightarrow2x>-6\\ \Leftrightarrow x>-3\)

\(b,\dfrac{x+2}{3}-\dfrac{3x-1}{5}< -2\\ \Leftrightarrow\dfrac{5\left(x+2\right)-3\left(3x-1\right)}{15}< -2\\ \Leftrightarrow5x+10-9x+3< -30\\ \Leftrightarrow-4x< -43\\ \Leftrightarrow x>\dfrac{43}{4}\)

\(a,\left(x-3\right)\left(x+3\right)-x\left(x+2\right)\le0\\ x^2-9-x^2-2x\le0\\ \Leftrightarrow-2x\le9\\ \Leftrightarrow x\ge-\dfrac{9}{2}\)

\(b,\dfrac{x-1}{4}-1>\dfrac{x+1}{3}+8\\ \Leftrightarrow\dfrac{3\left(x-1\right)-12-4\left(x+1\right)-8.12}{12}>0\\ \Leftrightarrow3x-3-12-4x-4-96>0\)

\(\Leftrightarrow-x>115\\ \Leftrightarrow x< -115\)

Answer 4

a) \(\left(x+3\right)\left(x-3\right)-x\left(x+2\right)\le0\)

\(\Leftrightarrow x^2-9-x^2-2x\le0\)

\(\Leftrightarrow-9-2x\le0\)

\(\Leftrightarrow2x\ge-9\)

\(\Leftrightarrow x\le-\dfrac{9}{2}\)

b) \(\dfrac{x-1}{4}-1>\dfrac{x+1}{3}+8\)

\(\Leftrightarrow\dfrac{x-5}{4}>\dfrac{x+25}{3}\)

\(\Leftrightarrow\dfrac{3\left(x-5\right)}{12}>\dfrac{4\left(x+25\right)}{12}\)

\(\Leftrightarrow3x-15>4x+100\)

\(\Leftrightarrow x< -115\)