Section 1.7
Parentheses and Minus Signs
A subtraction sign preceding a grouped expression must be treated very carefully.
Examples:
a. \begin{align}
5-(x+3)&=5-x-3&&\text{distribute the minus sign}
\end{align}
b. \begin{align}
5-(x+3)(x-2)&=5-(x^2+x-6)&&\text{multiply expressions in parenthesis}\\
&=5-x^2-x+6&&\text{distribute the minus sign}\\
&=11-x^2-x&&\text{combine like terms}
\end{align}
c. \begin{align}
\frac{1}{2}-\frac{x+2}{4}&=\frac{2}{4}-\frac{x+2}{4}&&\text{common denominator}\\
&=\frac{2-(x+2)}{4}&&\text{add fractions}\\
&=\frac{2-x-2}{4}&&\text{distribute the minus sign}\\
&=\frac{-x}{4}&&\text{combine like terms}
\end{align}
Exercises:
Simplify each of the following.
\begin{array}{l l}
1. \;\;1-2(x-1)(2x+1)\;\;& 2. \;\;2(x-3)-7(x+1)\;\;\\
3. \;\;{{(x+4)}^{2}}-{{(x-4)}^{2}}\;\;& 4. \;\;{{(3x-2)}^{2}}-{{(x+3)}^{2}}\;\;\\
5. \;\;3{{(2x-1)}^{2}}-2{{(x-3)}^{2}}\;\;& 6. \;\;(m+3)(m-1)-{{(x-2)}^{2}}+4\;\;\\
7. \;\;{{a}^{2}}-\left[ x(a-x)-a(x-a) \right]-{{a}^{2}}\;\;& 8. \;\;9p-(p+1)(p+2)-{{(p+3)}^{2}}\;\;\\
9. \;\;-7t-\left[ 5-3\left( t-2[1-4t] \right) \right]\;\;& 10. \;\;13a-\left[ -(7a+5)-\left( 6+2(8a-1)+3a \right) \right]\;\;\\
11. \;\;1-2\left( 2+3{{(x-1)}^{2}}+2(2x-1) \right)\;\;& 12. \;\;{{(2x-1)}^{2}}-2(x-2)(2x-1)\;\;\\
13. \;\;16x-2\left[ 8-2{{(2x-1)}^{2}} \right]\;\;& 14. \;\;3{{x}^{2}}-\left[ x-2(x+3)(x-2) \right]\;\;\\
15. \;\;1-2\left( 2-3{{(x-1)}^{2}}-2(2x-1) \right)\;\;& 16. \;\;{{(3x-1)}^{2}}-4(x-1)(x-2)-5{{x}^{2}}\;\;\\
17. \;\;x-\left\{ 2x-\left[ 3-4\left( x-5 \right) \right] \right\}\;\;& 18. \;\;x-\left\{ 2x\left[ 3-4\left( 5-x \right) \right] \right\}\;\;\\
19. \;\;2x-\left( 3x-5[2-x]-4[x-1] \right)\;\;& 20. \;\;{{x}^{2}}(2x-3)-x\left( {{x}^{2}}-4 \right)-x{{(x-1)}^{2}}\;\;
\end{array}
Write each fraction with the same denominator and then combine.
\begin{array}{l l}
21. \;\;\dfrac{3x}{6}-\dfrac{x-2}{2}\;\;& 22. \;\;\dfrac{2x-5}{6}-\dfrac{3x-5}{9}\;\;\\
23. \;\;5-\dfrac{x-3}{3}\;\;& 24. \;\;2-\dfrac{3x-5}{5}\;\;\\
25. \;\;x-\dfrac{5x+6}{5}\;\;& 26. \;\;\dfrac{6x+5}{6}-\dfrac{5x-6}{5}\;\;\\
27. \;\;\dfrac{2y-3}{3}-\dfrac{7-y}{4}\;\;& 28. \;\;\dfrac{2x+7}{2}-\dfrac{x+5}{3}-\dfrac{4x+11}{6}\;\;\\
29. \;\;\dfrac{2x-5}{3}-\dfrac{2x+5}{4}-\dfrac{x-10}{6}\;\;& 30. \;\;\dfrac{x-7}{2}-\dfrac{x+7}{4}-\dfrac{4x-3}{6}\;\;
\end{array}