we know that
The easiest way to find the number that makes the statement true is to substitute the given answers into the equation.
so
Part 1)
[tex]4.3-2x < x^{2} -2[/tex]
a) For [tex]x=0[/tex]
[tex]4.3-2*0 < 0^{2} -2[/tex]
[tex]12 < -2[/tex] -----> is false
b) For [tex]x=1[/tex]
[tex]4.3-2*1 < 1^{2} -2[/tex]
[tex]10 < -1[/tex] -----> is false
c) For [tex]x=2[/tex]
[tex]4.3-2*2 < 2^{2} -2[/tex]
[tex]8 < 2[/tex] -----> is false
d) For [tex]x=3[/tex]
[tex]4.3-2*3 < 3^{2} -2[/tex]
[tex]6 < 7[/tex] -----> is true
therefore
the answer part 1) is the option D
[tex]x=3[/tex]
Part 2)
[tex]6*7-5x+x^{2} \geq x^{3} +x^{2}[/tex]
a) For [tex]x=5[/tex]
[tex]6*7-5*5+5^{2} \geq 5^{3} +5^{2}[/tex]
[tex]42 \geq 150[/tex] ----> is false
b) For [tex]x=6[/tex]
[tex]6*7-5*6+6^{2} \geq 6^{3} +6^{2}[/tex]
[tex]48 \geq 252[/tex] ----> is false
c) For [tex]x=4[/tex]
[tex]6*7-5*4+4^{2} \geq 4^{3} +4^{2}[/tex]
[tex]38 \geq 80[/tex] ----> is false
d) For [tex]x=3[/tex]
[tex]6*7-5*3+3^{2} \geq 3^{3} +3^{2}[/tex]
[tex]36 \geq 36[/tex] ----> is True
therefore
the answer Part 2) is the option D
[tex]x=3[/tex]
Part 3)
we have
[tex]8*3-4x\ (\ )\ 7*(10-x)[/tex]
For [tex]x=3[/tex]
[tex]8*3-4*3\ (\ )\ 7*(10-3)[/tex]
[tex]12\ (\ )\ 49[/tex]
we know that
[tex]12 < 49[/tex]
therefore
the answer Part 3) is the option B
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Part 4)
we have
[tex]9*4-2x\ (\ )\ 5*(11-x)[/tex]
For [tex]x=8[/tex]
[tex]9*4-2*8\ (\ )\ 5*(11-8)[/tex]
[tex]20\ (\ )\ 15[/tex]
we know that
[tex]20 > 15[/tex]
therefore
the answer Part 4) is the option B
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