Inequalities

1. Overview

Inequalities are mathematical statements that compare two values or expressions using symbols like $<, >, \le,$ or $\ge$. Unlike equations that have a single solution, inequalities typically have a range of solutions. This topic is crucial for problems involving constraints or limits, and for understanding solution sets on number lines and graphs. You'll need to master solving inequalities algebraically and representing them graphically, especially in the extended curriculum.

Key Definitions

• Inequality: A mathematical sentence that uses symbols like $<, >, \le,$ or $\ge$ to compare two values.
• Solution Set: The range of all possible values that satisfy an inequality.
• Strict Inequality: Inequalities using $<$ (less than) or $>$ (greater than) where the boundary value is not included.
• Non-strict Inequality: Inequalities using $\le$ (less than or equal to) or $\ge$ (greater than or equal to) where the boundary value is included.
• Integer: A whole number (positive, negative, or zero). Exam questions often ask for "integer solutions."

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Core Content

Representing Inequalities on a Number Line

When showing a range of values on a number line:

• Use an open circle (○) for strict inequalities ($<$ or $>$).
• Use a closed (solid) circle (●) for non-strict inequalities ($\le$ or $\ge$).
• Draw a line with an arrow to show the direction of the solution.

Worked example 1 — Single Inequality

Represent $x > -2$ on a number line.

1. Locate $-2$ on the number line.
2. Draw an open circle (○) at $-2$ because the inequality is "greater than" but not equal to.
3. Draw a line pointing to the right (towards larger numbers) from the open circle.

Worked example 2 — Combined Inequality

Represent $-1 \le x < 3$ on a number line.

1. Locate $-1$ and $3$ on the number line.
2. Place a solid circle (●) at $-1$ because the inequality includes "equal to" ($\le$).
3. Place an open circle (○) at $3$ because the inequality is strictly "less than" ($<$).
4. Join the two circles with a solid line to show that $x$ is between these values.

Worked example 3 — Inequality with Integer Constraint

Represent the integer values of $x$ such that $-3 < x \le 2$ on a number line.

1. Identify the integers that satisfy the inequality: $-2, -1, 0, 1, 2$.
2. Locate these integers on the number line.
3. Place a solid circle (●) at each integer value.

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Extended Content (Extended Curriculum Only)

Solving Linear Inequalities

Solving an inequality is very similar to solving an equation. You perform the same operations on both sides to isolate the variable.

The Golden Rule: When you multiply or divide both sides by a negative number, you must flip the inequality sign.

Worked example 4 — Solving Algebraically

Solve: $10 - 3x \le 22$

1. Subtract $10$ from both sides: $10 - 3x - 10 \le 22 - 10$ $-3x \le 12$ Reason: Isolate the term with $x$
2. Divide both sides by $-3$. Caution: Since we are dividing by a negative, flip the $\le$ to $\ge$: $\frac{-3x}{-3} \ge \frac{12}{-3}$ Reason: Isolate $x$, and flip the sign because we divided by a negative number.
3. Final Answer: $x \ge -4$

Worked example 5 — Solving with Fractions

Solve: $\frac{2x + 1}{3} > 5$

1. Multiply both sides by $3$: $3 \cdot \frac{2x + 1}{3} > 5 \cdot 3$ $2x + 1 > 15$ Reason: Eliminate the fraction.
2. Subtract $1$ from both sides: $2x + 1 - 1 > 15 - 1$ $2x > 14$ Reason: Isolate the term with $x$.
3. Divide both sides by $2$: $\frac{2x}{2} > \frac{14}{2}$ $x > 7$ Reason: Isolate $x$.
4. Final Answer: $x > 7$

Graphical Representation in Two Variables

This involves shading regions on a Cartesian $(x, y)$ graph.

1. Replace the inequality sign with $=$ and draw the line.
2. Solid line: Used for $\le$ or $\ge$ (the boundary line is included in the solution).
3. Dashed (broken) line: Used for $<$ or $>$ (the boundary line is not included in the solution).
4. Test a point: Choose a point not on the line (usually $(0,0)$ if the line doesn't pass through the origin). Plug the coordinates into the inequality.
   • If the statement is true, shade the side containing $(0,0)$.
   • If the statement is false, shade the other side.
   • Note: Always check the question instructions; sometimes you are asked to shade the "unwanted" region.

Worked example 6 — Identifying a Region

Identify the region satisfied by $y > x + 1$.

1. Draw the line $y = x + 1$. Since the sign is $>$, use a dashed line.
2. Test point $(0,0)$: $0 > 0 + 1 \Rightarrow 0 > 1$ (This is False). Reason: Substitute $x=0$ and $y=0$ into the inequality.
3. Since $(0,0)$ makes the inequality false, the solution is the region on the other side of the line (the region above the line). Shade the region above the dashed line.

Worked example 7 — Shading the Unwanted Region

Shade the unwanted region for the inequality $y \le 2x - 1$.

1. Draw the line $y = 2x - 1$. Since the sign is $\le$, use a solid line.
2. Test point $(0,0)$: $0 \le 2(0) - 1 \Rightarrow 0 \le -1$ (This is False). Reason: Substitute $x=0$ and $y=0$ into the inequality.
3. Since $(0,0)$ makes the inequality false, the unwanted region is the region containing $(0,0)$, which is above the line. Shade the region above the solid line. The solution is the unshaded region below the line.

Listing Inequalities for a Given Region

To find the inequalities that define a shaded region:

1. Find the equation of each boundary line ($y = mx + c$, $x = k$, or $y = k$).
2. Determine if the line is solid or dashed.
3. Test a point inside the unshaded region to see which inequality sign ($<, >, \le, \ge$) makes the statement true.

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Key Equations

There are no specific formulas provided on the IGCSE formula sheet for inequalities. You must memorise the "flip" rule for negatives and the general form of a linear equation:

• Linear form: $y = mx + c$ (where $m$ is gradient and $c$ is y-intercept).
• Vertical lines: $x = k$
• Horizontal lines: $y = k$

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Common Mistakes to Avoid

• ❌ The Negative Flip: Forgetting to reverse the inequality sign when multiplying or dividing by a negative number.
  • Example: $-2x < 10$ ❌ $x < -5$ (Wrong) ✓ $x > -5$ (Correct)
• ❌ Line Styles: Using a solid line for strict inequalities ($<$ or $>$) on a graph. This indicates that the boundary is included in the solution, which is incorrect.
  • Example: For $y > x + 1$, drawing a solid line implies $y \ge x + 1$.
• ❌ Zero Error: Assuming $3 > \frac{1}{x}$ means $3x > 1$ without considering the sign of $x$. If $x$ is negative, the inequality sign must flip.
  • Example: If $x = -1$, then $3 > \frac{1}{-1}$ is $3 > -1$ (True). But $3x > 1$ becomes $-3 > 1$ (False).
• ❌ Defaulting to Equals: In word problems, writing $x = 5$ when the context requires $x \ge 5$ (e.g., "at least 5"). Always carefully consider the wording of the problem.
  • Example: "The minimum number of tickets John must sell is 5" translates to $x \ge 5$, not $x = 5$.
• ❌ Integer Solutions: Not listing all integer solutions when asked. For example, if $x > -2$ and $x < 3$ and $x$ is an integer, the solution is $-1, 0, 1, 2$, not just $x > -2$ and $x < 3$.

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Exam Tips

• Read the "Shading" instruction carefully: IGCSE papers often ask you to "Shade the unwanted region" so that the correct solution set is left white. If you shade the correct region by mistake, you may lose all marks for that section.
• Command Word "List": If a question says "List the integers that satisfy...", do not just give the inequality. Write out the actual numbers (e.g., $1, 2, 3$).
• Calculator Use: You can use your calculator to check coordinates on a boundary line, but you must show the algebraic steps for solving the inequality to get full "Method" marks.
• Integer Boundaries: If $x < 5$ and $x$ must be an integer, the largest possible value is $4$. If $x \le 5$, the largest value is $5$. Pay close attention to the "or equal to" bar!
• Check your answer: After solving an inequality, substitute a value from your solution set back into the original inequality to verify that it holds true. This can help you catch errors, especially with the "flip the sign" rule.