Solving Inequalities: Master the Method and Avoid Common Mistakes

What the examiner is testing

The examiner is assessing your ability to find the range of values that satisfy an inequality, rather than a single specific value. Marks are awarded for correctly manipulating the inequality while preserving its direction and for presenting the final answer in the correct format.

The method

1. Isolate the variable term: Use inverse operations (addition, subtraction) to get all terms containing the variable on one side of the inequality and all constant terms on the other.
2. Simplify the variable term: Divide or multiply both sides by the coefficient of the variable to make the variable stand alone.
3. Flip the inequality sign if multiplying or dividing by a negative number: This is the crucial step unique to inequalities. If you multiply or divide both sides by a negative number, the direction of the inequality sign must reverse.
4. Write the solution: Present the final range of values for the variable.

Worked example

Solve \(3x - 5 < 10\).

$$ \begin{aligned} 3x - 5 &< 10 \\ 3x - 5 + 5 &< 10 + 5 \\ 3x &< 15 \\ \frac{3x}{3} &< \frac{15}{3} \\ x &< 5 \end{aligned} $$

Sanity check: If \(x=4\) (which is less than 5), then \(3(4) - 5 = 12 - 5 = 7\). \(7 < 10\), which is true. If \(x=6\) (which is not less than 5), then \(3(6) - 5 = 18 - 5 = 13\). \(13 < 10\), which is false. The solution \(x < 5\) is correct.

Worked example: a harder one

Solve \(7 - 2x \ge 15\).

A common first move might be to subtract 7 from both sides:
$$ \begin{aligned} 7 - 2x &\ge 15 \\ -2x &\ge 8 \end{aligned} $$
Now, to isolate \(x\), we must divide by -2. This is where the crucial rule applies.

$$ \begin{aligned} \frac{-2x}{-2} &\le \frac{8}{-2} \quad \text{(The inequality sign flips!)} \\ x &\le -4 \end{aligned} $$

Alternatively, you could add \(2x\) to both sides first to avoid dividing by a negative:
$$ \begin{aligned} 7 - 2x &\ge 15 \\ 7 &\ge 15 + 2x \\ 7 - 15 &\ge 2x \\ -8 &\ge 2x \\ \frac{-8}{2} &\ge \frac{2x}{2} \\ -4 &\ge x \end{aligned} $$
Both methods lead to the same correct solution, \(x \le -4\).

Practice

1. Solve \(4x + 7 \le 23\)
2. Solve \(12 - x > 5\)
3. Solve \(5(x - 3) < 2x + 9\)
4. Find the integer values of \(n\) that satisfy \(2n - 1 \le 11\) and \(3n + 2 > -7\).

Answers:

1. \(x \le 4\)
2. \(x < 7\)
3. \(x < 8\)
4. Working for question 4:
   First inequality:
   $$ \begin{aligned} 2n - 1 &\le 11 \\ 2n &\le 12 \\ n &\le 6 \end{aligned} $$
   Second inequality:
   $$ \begin{aligned} 3n + 2 &> -7 \\ 3n &> -9 \\ n &> -3 \end{aligned} $$
   Combining these, we need \(n > -3\) and \(n \le 6\).
   The integer values of \(n\) are \(-2, -1, 0, 1, 2, 3, 4, 5, 6\).

The three mistakes that lose marks

1. Forgetting to flip the inequality sign: This is the most common error. If you divide or multiply both sides by a negative number, the sign must reverse.
   • Wrong answer: Solving \(-3x > 9\) as \(x > -3\).
   • Correct answer: \(x < -3\).
2. Incorrectly combining multiple inequalities: When you have two inequalities, you need to find the values that satisfy both conditions. Do not simply write the two separate solutions next to each other without considering the overlap.
   • Wrong answer: For \(x > 2\) and \(x < 5\), writing " \(x > 2\) and \(x < 5\) ".
   • Correct answer: \(2 < x < 5\).
3. Errors with integer solutions: If the question asks for integer solutions, ensure your final list only includes whole numbers within the calculated range.
   • Wrong answer: For \(2 < x \le 5\), listing the integers as \(2, 3, 4, 5\).
   • Correct answer: \(3, 4, 5\) (since \(x\) must be greater than 2).

30-second recap

Treat inequalities like equations, but remember to flip the sign if you multiply or divide by a negative number. When solving two inequalities, combine their solutions to find the common range. Pay close attention to whether the question asks for a range or specific integer values.