3. Tough Linear Inequalities

Harder linear inequalities often include parentheses on both sides, negative coefficients, or variables on both sides.

Example 1:
\[ 4(2x-3)+5\ge37. \]
Distribute and combine:
\[ 8x-12+5\ge37,\qquad 8x-7\ge37. \]
Add 7 and divide:
\[ 8x\ge44,\qquad x\ge\frac{11}{2}. \]

Example 2:
\[ 8-2x\le x-13. \]
Add \(2x\):
\[ 8\le3x-13. \]
Add 13:
\[ 21\le3x. \]
Divide:
\[ 7\le x, \]
which means \(x\ge7\).

Example 3:
\[ 5-(2x-7)>18. \]
Distribute the negative:
\[ 5-2x+7>18. \]
Combine:
\[ 12-2x>18. \]
Subtract 12:
\[ -2x>6. \]
Divide by \(-2\), so flip:
\[ x<-3. \]

Common mistake: rewriting \(7\le x\) as \(x\le7\). If 7 is less than or equal to \(x\), then \(x\) is greater than or equal to 7.