Solving inequalities
The process to solve inequalities is the same as the process to solve equations, which uses Inverse operations are opposite calculations often used in solving equations. To remove +9 from a sum, perform the inverse operation which is -9. to keep the equation or inequality balanced. Instead of using an equals sign, however, the inequality symbol is used throughout.
Example
Solve the inequality \(3m + 2 \textgreater -4\).
The inequality will be solved when \(m\) is isolated on one side of the inequality. This can be done by using inverse operations at each stage of the process.
\(\begin{array}{rcl} 3m + 2 & \textgreater & -4 \\ -2 && -2 \\ 3m & \textgreater & -6 \\ \div 3 && \div 3 \\ m & \textgreater & -2 \end{array}\)
The final answer is \(m \textgreater -2\), which means \(m\) can be any value that is bigger than -2, not including -2 itself. If this answer was to be placed on a number line, an open circle would be needed at -2 with a line indicating the numbers that are greater than 2.
Question
Solve the inequality \(2(2c + 2) \leq 5\). Show the answer on a number line.
\(2(2c + 2) \leq 5\)
Expand the bracket:
\(4c + 4 \leq 5\)
Subtract 4 from each side:
\(4c \leq 1\)
Divide each side by 4:
\(c \leq \frac{1}{4}\)
\(c \leq \frac{1}{4}\) is the final answer.
To show this answer on a number line, put a closed circle at \(\frac{1}{4}\) and indicate the numbers below this point.
Extra care should be taken when calculating with a negative unknown. Use an inverse operation to make the coefficient of the unknown positive.
Question
Solve the inequality \(3(3 - x) \textless 6\). Show the answer on a number line.
\(3(3 - x) \textless 6\)
Expand the bracket:
\(9 - 3x \textless 6\)
The unknown has a coefficient of -3, so now add \(3x\) to both sides of the inequality:
\(9 \textless 3x +6\)
Subtract 6 from both sides:
\(3 \textless 3x\)
Now divide both sides by 3:
\(1 \textless x\), which can also be written as \(x \textgreater 1\) (reading from right to left).
To show this answer on a number line, put an open circle at 1 and indicate the numbers greater than 1.
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