For Problems 1 and 2, write the algebraic expression.
a. Twenty-five less than three times a number.
b. A number increased by eighteen.
a. The difference between twice a number and six.
b. A number divided by three.
For Problems 3 to 5, simplify the algebraic terms, and express the answer with a positive exponent.
a. [latex] (-x)^3(-x)^4 [/latex]
b. [latex] \displaystyle{\frac{(-x)^{-5}(-x)^3}{(-x)^{-4}}} [/latex]
a. [latex] \displaystyle{(x^{-6})^{\frac{1}{3}}} [/latex]
b. [latex] \sqrt[3]{x^6} [/latex]
a. [latex] \displaystyle{(16x^0y^4)^{\frac{1}{2}}} [/latex]
b. [latex] (-2x^{-2}y^{-4})^{-1} (2x^{-2})^2 [/latex]
For Problems 6 and 7, simplify the expressions, then evaluate for the given value of the variables in the brackets.
a. [latex] -3x^2 + 2x + 2x^2 - 8x + 10 [/latex] [latex] (x = -3) [/latex]
b. [latex] 5(2x - 3y) - 2(3x - 2y) + 7 [/latex] [latex] (x = 2 and y = 1) [/latex]
a. [latex] (x + 3)^2 - (x + 2)(x - 2) [/latex] [latex] (x = 3) [/latex]
b. [latex] (2x + 5)^2 - (3x - 1 )^2 [/latex] [latex] (x = 1) [/latex]
For Problems 8 to 11, factor the expressions.
a. [latex] 8xy^2 - 6x^2y [/latex]
b. [latex] 10ab - 8bc [/latex]
a. [latex] 4xy - 20y - x^2 + 5x [/latex]
b. [latex] 1 - 121x^2 [/latex]
a. [latex] 2x^2 - 22x + 56 [/latex]
b. [latex] 4x^2 - 9x - 9 [/latex]
a. [latex] x^2 + 10x + 25 [/latex]
b. [latex] x^2 + 6xy - 16y^2 [/latex]
For Problems 12 and 13, write the algebraic equation and solve.
a. Nine less than twice a number is twenty-one.
b. Twenty-two is five times a number less than three.
a. Four times eight is sixteen times a number.
b. Thirty is a product of six and a number.
For Problems 14 and 15, solve for the unknown variable, x, using the principles of equality.
a. [latex] 24 5x = 4 [/latex]
b. [latex] \displaystyle{\frac{x}{3} - 2 = 4} [/latex]
a. [latex] 8 + 2x = 4 5x [/latex]
b. [latex] b. 3(3x 3) = 33 [/latex]