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Solutions and answers:[tex] \\ [/tex]1.) [tex]\color {red} \boxed{ \bold{ x^2 + 3x - 10 \geq 0 }}[/tex]Factor the quadratic: [tex] \sf {(x+5)(x-2) \geq 0}[/tex]Find the critical points: [tex] \sf {x = -5, x = 2}[/tex]Test íntervals:[tex] \sf {x < -5}[/tex]: (-)(-) is positive, so [tex] \sf {x \leq -5}[/tex] is part of the solution.[tex] \sf {-5 < x < 2}[/tex]: (+)(-) is negative, so this interval is not part of the solution.[tex] \sf {x > 2}[/tex]: (+)(+) is positive, so x \geq 2 is part of the solution.Solution: [tex] \color {red} \boxed {\sf { x \leq -5 }}[/tex]or [tex] \color {red} \boxed { \sf {x \geq 2}}[/tex]______________________________2.) [tex] \color {orange} \boxed{\bold {x^2 - x \geq 12}}[/tex]Rearrange: [tex] \sf {x^2 - x - 12 \geq 0}[/tex]Factor the quadratic: [tex] \sf {(x-4)(x+3) \geq 0}[/tex]Find the critical points: [tex] \sf {x = 4, x = -3}[/tex]Test íntervals:[tex]\sf {x < -3: (-)(-) > 0}[/tex], so [tex] \sf {x \leq -3}[/tex] is part of the solution.[tex] \sf {-3 < x < 4}[/tex]: (-)(+) < 0, so this interval is not part of the solution.[tex] \sf {x > 4}[/tex]: (+)(+) > 0, so[tex] \sf { x \geq 4 }[/tex]is part of the solution.Solution: [tex] \color {orange} \boxed{\sf {x \leq -3}}[/tex] or [tex] \color {orange}\boxed{\sf {x \geq 4}}[/tex]______________________________3.) [tex] \color {yellow} \boxed{\bold { x^2 + 12x + 32 < 0}}[/tex]Factor the quadratic: [tex] \sf {(x+8)(x+4) < 0}[/tex]Find the critical points: [tex] \sf {x = -8, x = -4}[/tex]Test íntervals:[tex] \sf {x < -8}[/tex]: (-)(-) > 0, so this interval is not part of the solution.[tex] \sf {-8 < x < -4}[/tex]: (+)(-) < 0, so this interval is part of the solution.[tex] \sf {x > -4}[/tex]: (+)(+) > 0, so this interval is not part of the solution.Solution: [tex]\color {yellow}\boxed{ \sf {-8 < x < -4}}[/tex]______________________________4.) [tex] \color {green} \boxed{\bold { x^2 + 2x - 15 < 0}}[/tex]Factor the quadratic: [tex] \sf { (x+5)(x-3) < 0}[/tex]Find the critical points: [tex] \sf { x = -5, x = 3}[/tex]Test íntervals:[tex] \sf {x < -5}[/tex]: (-)(-) > 0, so this interval is not part of the solution.[tex] \sf {-5 < x < 3}[/tex]: (+)(-) < 0, so this interval is part of the solution.[tex] \sf {x > 3}[/tex]: (+)(+) > 0, so this interval is not part of the solution.Solution: [tex]\color {green} \boxed{\sf {-5 < x < 3}}[/tex]______________________________5.) [tex] \color {skyblue} \boxed{\bold {x(x+4) > 21}}[/tex]Expand and rearrange: [tex] \sf {x^2 + 4x - 21 > 0}[/tex]Factor the quadratic: [tex] \sf {(x+7)(x-3) > 0}[/tex]Find the critical points: [tex] \sf { x = -7, x = 3}[/tex]Test íntervals:[tex] \sf {x < -7}[/tex]: (-)(-) > 0, so x < -7 is part of the solution.[tex] \sf {-7 < x < 3}[/tex]: (+)(-) < 0, so this interval is not part of the solution.[tex] \sf {x > 3}[/tex]: (+)(+) > 0, so x > 3 is part of the solution.Solution: [tex]\color {skyblue}\boxed{ \sf { x < -7 }} [/tex] or [tex] \color {skyblue}\boxed{\sf{ x > 3}}[/tex]

Answered by PrincessUmbriel | 2025-08-12