This tutorial explains how to solve linear inequalities step by step. You will review the fundamental properties of inequalities and apply them to solve standard inequalities, double inequalities, and inequalities containing fractional expressions.
Let \( a \), \( b \), and \( c \) be real numbers.
These properties also hold when:
Solve the inequality:
\[ 6x - 6 > 2x + 2 \]Add 6 to both sides:
\[ 6x > 2x + 8 \]Subtract \( 2x \) from both sides:
\[ 4x > 8 \]Divide both sides by \( 4 \):
\[ x > 2 \]Conclusion
\[ (2, +\infty) \]Matched Exercise
\[ 10x - 8 > 4x + 10 \]Solve the inequality:
\[ 2(3x + 2) - 20 > 8(x - 3) \]Add 16 to both sides:
\[ 6x > 8x - 8 \]Subtract \( 8x \) from both sides:
\[ -2x > -8 \]Divide both sides by \( -2 \) and reverse the inequality sign:
\[ x < 4 \]Conclusion
\[ (-\infty, 4) \]Matched Exercise
\[ -3(4x + 1) + 10 > -4(x - 3) \]Solve:
\[ -3 < 4(x + 2) - 3 < 9 \]Subtract 5 from all three parts:
\[ -8 < 4x < 4 \]Divide all parts by 4:
\[ -2 < x < 1 \]Conclusion
\[ (-2, 1) \]Matched Exercise
\[ -1 < -2(x - 3) - 3 < 7 \]Solve:
\[ \frac{x + 2}{3} - \frac{2}{5} < \frac{-x - 1}{3} - \frac{1}{6} \]Multiply all terms by the least common denominator \( 30 \):
\[ 10(x + 2) - 12 < 10(-x - 1) - 5 \] \[ 10x + 20 - 12 < -10x - 10 - 5 \] \[ 10x + 8 < -10x - 15 \]Add \( 10x \) to both sides:
\[ 20x < -23 \]Divide both sides by 20:
\[ x < -\frac{23}{20} \]Conclusion
\[ (-\infty, -\frac{23}{20}) \]Matched Exercise
\[ \frac{x - 2}{4} - \frac{2}{7} < \frac{-x + 3}{7} - \frac{1}{2} \]