7. Review Exercises

Answers to the odd-numbered questions are available at the end of the book.

For problems 1 to 4, write the algebraic expression.

1. a. Twelve increased by three times a number.
   b. The difference between a number and five.
2. a. Eight decreased by twice a number.
   b. Six less than the total of a number and ten.
3. a. The product of three more than a number and the number.
   b. Sum of ten times a number and fifteen.
4. a. Sum of fifteen and half of a number.
   b. Product of two times a number and seven.

For problems 5 to 10, simplify the expressions, then evaluate for the given value of the variables in the brackets.

5. a. [latex]-4x^2 + 3x - 5 + 7x^2 - 2x + 3[/latex]   [latex](x = 2)[/latex]
   b. [latex]4x^2 - 5 + 7x - 2x^2 - x - 3[/latex]   [latex](x = -1)[/latex]
6. a. [latex]3x^2 - x + 2 + x^2 - 5x - 2[/latex]   [latex](x = 3)[/latex]
   b. [latex]-5y^2 - 7y + 3 + y^2 - 5y + 2[/latex]   [latex](y = -2)[/latex]
7. a. [latex]-y^2 + 4xy + x^2 - 6y^2 - xy - 11x^2[/latex]   [latex](x = 1, y = 2)[/latex]
   b. [latex](x - 4)(x + 2) + 3(x + 2)[/latex]   [latex](x = 3)[/latex]
8. a. [latex]-4x^2 + 6xy - 6y^2 + 6x^2 - 2xy + 3y^2[/latex]   [latex](x = 2, y = 1)[/latex]
   b. [latex](y - 2)(y - 3) + 2(y - 2)[/latex]   [latex](y = 4)[/latex]
9. a. [latex](2x - 3)^2 - (x + 3)^2[/latex]   [latex](x = 4)[/latex]
   b. [latex](5 + x)^2 + (4 - x)(4 + x)[/latex]   [latex](x = 5)[/latex]
10. a. [latex](2x + 1)^2 - (x - 2)^2[/latex]   [latex](x = 1)[/latex]
    b. [latex](3 - x)^2 + (x - 3)(x + 3)[/latex]   [latex](x = 2)[/latex]

For problems 11 to 14, write the algebraic equation and solve.

11. a. Seventeen more than five times a number is forty-two.
    b. A number divided by fifteen is forty-five.
12. a. The product of five and a number is seventy-five.
    b. Three more than two times a number is nine.
13. a. The difference between a number and ten is ten
    b. The product of four times a number and three is thirty-six.
14. a. The sum of two times a number and eight is one hundred.
    b. A number divided by three is seven.

For problems 15 to 18, solve for the unknown variable, x, using the principles of equality.

15. a. [latex]5x - 5 = 10[/latex]
    b. [latex]\displaystyle{\frac{x}{3} + 4 = 10}[/latex]
16. a. [latex]3x - 5 = -17[/latex]
    b. [latex]\displaystyle{\frac{x}{4} - 2 = 1}[/latex]
17. a. [latex]12 - 3x = 3 - 4x[/latex]
    b. [latex]4(x + 4) = 24[/latex]
18. a. [latex]4x - 2 = 13 - 6x[/latex]
    b. [latex]3(2x - 5) = 3[/latex]