To solve the polynomial inequality ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905377/gif.latex "x^{3} + 7x^{2} < 4x+ 28") it is necessary to get it into standard form by writing all terms at the same side. Subtract ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905378/gif.latex "4x+ 28") from both sides: ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905379/gif.latex "x^{3} + 7x^{2} - (4x+ 28) < 4x+ 28 - (4x+ 28)") ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905380/gif.latex "x^{3} + 7x^{2} - 4x-28 < 0") The values of ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905381/gif.latex "x") that serve as boundaries of the solutions set are the zeroes of the polynomial, so solve for ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905382/gif.latex "x") in the equation: ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905383/gif.latex "x^{3} + 7x^{2} - 4x-28 = 0") Factor the polynomial by grouping and taking out the GCFs: ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905384/gif.latex "(x^{3} + 7x^{2}) - (4x+28) = 0") ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905385/gif.latex "x^{2} (x + 7) - 4(x + 7) = 0") ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905386/gif.latex "(x^{2} - 4)(x + 7) = 0") ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905387/gif.latex "x^{2} - 4") is the difference of squares and can be factored according to a pattern: ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905388/gif.latex "(x^{2} - 2^{2})(x + 7) = 0") ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905389/gif.latex "(x+2)(x-2)(x+7) = 0") By the Zero Factor Property, one of the binomials is equal to 0, so ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905390/gif.latex "x-2 = 0") in which case ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905391/gif.latex "x = 2"), ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905392/gif.latex "x+2 = 0") in which case ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905393/gif.latex "x = -2"), or ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905394/gif.latex "x+7 = 0") in which case ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905395/gif.latex "x = -7"). ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905396/gif.latex "-7"), ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905397/gif.latex "-2"), and 2 are the boundaries of the intervals, which are ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905398/gif.latex "(-\infty, -7)"), ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905399/gif.latex "(-7, -2)"), ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905400/gif.latex "(-2,2)"), and ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905401/gif.latex "(2, \infty)"). To determine which intervals belong to the solution set, choose one value from each, substitute for ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905402/gif.latex "x") in the original inequality, and determine whether it is true. ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905403/gif.latex "(-\infty, -7)"): Choose ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905404/gif.latex "x = -8") ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905405/gif.latex "x^{3} + 7x^{2} < 4x+ 28") ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905406/gif.latex "(-8)^{3} + 7 \cdot (-8)^{2} < 4(-8)+ 28") ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905407/gif.latex "-512 + 7 \cdot 64 < -32+ 28") ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905408/gif.latex "-512 + 448 < -32+ 28") ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905409/gif.latex "-64 < -4") True - include ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905410/gif.latex "(-\infty, -7)") ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905411/gif.latex "(-7, -2)"): Choose ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905412/gif.latex "x = -3") ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905413/gif.latex "x^{3} + 7x^{2} < 4x+ 28") ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905414/gif.latex "(-3)^{3} + 7 \cdot (-3)^{2} < 4(-3)+ 28") ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905415/gif.latex "-27 + 7 \cdot 9 < -12 + 28") ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905416/gif.latex "-27 + 63 < -12 + 28") ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905417/gif.latex "36 < 16") False - do not include ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905418/gif.latex "(-7, -2)") ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905419/gif.latex "(-2,2)"): Choose ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905420/gif.latex "x= 0") ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905421/gif.latex "x^{3} + 7x^{2} < 4x+ 28") ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905422/gif.latex "0^{3} + 7 \cdot 0^{2} < 4 \cdot 0 + 28") ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905423/gif.latex "0+0 < 0 + 28") ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905424/gif.latex "0 < 28") True - include ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905425/gif.latex "(-2,2)") ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905426/gif.latex "(2, \infty)"): Choose ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905427/gif.latex "x = 3") ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905428/gif.latex "x^{3} + 7x^{2} < 4x+ 28") ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905429/gif.latex "3^{3} + 7 \cdot 3^{2} < 4 \cdot 3+ 28") ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905430/gif.latex "27 + 7 \cdot 9 < 12+ 28") ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905431/gif.latex "27 + 63 < 12+ 28") ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905432/gif.latex "90 < 40") False - do not include ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905433/gif.latex "(2, \infty)") Also, the boundary values themselves should not be included, because the inequality symbol ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905434/gif.latex "<") does not allow for equality. The solution set is ![](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/905435/gif.latex "\left \{ x| x < -7 \textrm{ or } -2 < x < 2 \right \}")

Diagnostic Test 5

Solve for : (Nearest hundredth, if applicable).

Solve for :

$3^{2x+ 1} = 9^{2x- 4 }$

(Nearest hundredth, if applicable).