Algebra Revision Notes

Basic Inequality Symbols

Inequalities use specific symbols to compare values. Understand these symbols thoroughly:

• $a < b$: $a$ is less than $b$.
• $a > b$: $a$ is greater than $b$.
• $a \le b$: $a$ is less than or equal to $b$.
• $a \ge b$: $a$ is greater than or equal to $b$.

These symbols define a range of possible values for a variable rather than a single specific value, as in an equation.

Solving Linear Inequalities

The process for solving linear inequalities is very similar to solving linear equations. You can add or subtract the same number from both sides, and multiply or divide both sides by a positive number without changing the inequality direction. The goal is to isolate the variable on one side. Always perform operations on both sides to maintain balance, just like with equations. For example, to solve $x - 5 < 2$, add 5 to both sides: $x < 7$.

The Key Rule for Negative Multiplication/Division

This is the most critical difference between solving equations and inequalities. If you multiply or divide both sides of an inequality by a negative number, you MUST reverse the direction of the inequality sign. For example, if you have $-2x > 6$, dividing by $-2$ means $x < -3$. Failing to reverse the sign will result in an incorrect solution set. Remember this rule!

Representing Solutions

Solutions to inequalities can be written in inequality notation (e.g., $x < 5$) or represented on a number line. On a number line:

• An open circle ($o$) is used for strict inequalities ($<$ or $>$) to show that the endpoint is not included.
• A closed circle ($\bullet$) is used for non-strict inequalities ($\le$ or $\ge$) to show that the endpoint is included. A line is then drawn from the circle in the direction indicated by the inequality.

What are Inequalities?

An inequality compares two expressions using one of four symbols: < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). For example, 'x > 5' means 'x is greater than 5'. Inequalities represent a range of possible values for a variable, rather than a single specific value like in an equation. It's crucial to distinguish these symbols to correctly interpret the conditions they impose.

Solving Inequalities - Basic Operations

The general rules for solving inequalities are similar to solving equations. You can add or subtract the same number from both sides of the inequality, and the inequality symbol remains unchanged. For example, if x - 3 < 7, then adding 3 to both sides gives x < 10. Similarly, you can multiply or divide both sides by the same positive number without changing the inequality direction. If 2x ≥ 8, dividing by 2 gives x ≥ 4.

Solving Inequalities - The Crucial Negative Rule

The most important rule when solving inequalities is that if you multiply or divide both sides of an inequality by a negative number, you MUST reverse the direction of the inequality symbol. For instance, if you have -3x < 12, dividing both sides by -3 means you change the < to >, resulting in x > -4. Failure to flip the symbol when multiplying or dividing by a negative number is a common mistake and will lead to an incorrect solution.

Representing Solutions on a Number Line

Once an inequality is solved, its solution can be visually represented on a number line. Use a hollow circle (┒) for strict inequalities (< or >) to show that the endpoint value is NOT included. Use a solid (filled) circle (●) for inclusive inequalities (≤ or ≥) to show that the endpoint value IS included. An arrow extending from the circle indicates the range of values that satisfy the inequality. For example, x > 2 would be a hollow circle at 2 with an arrow pointing right. x ≤ -1 would be a solid circle at -1 with an arrow pointing left.

Solving Linear Inequalities

Linear inequalities are solved much like linear equations. The goal is to isolate the variable. You can add or subtract the same value from both sides, and multiply or divide both sides by a positive value without changing the inequality sign. However, if you multiply or divide both sides by a negative number, you MUST reverse the direction of the inequality sign. For example, if $-2x < 6$, dividing by $-2$ gives $x > -3$. Always check your solution by substituting a value from your solution set back into the original inequality.

Solving Quadratic Inequalities

To solve quadratic inequalities (e.g., $ax^2 + bx + c > 0$), first arrange the inequality so that one side is zero. Then, find the roots of the corresponding quadratic equation $ax^2 + bx + c = 0$ by factoring, using the quadratic formula, or completing the square. These roots divide the number line into intervals. Test a value from each interval in the original inequality to determine which intervals satisfy it. Alternatively, make a sketch of the quadratic graph $y = ax^2 + bx + c$. The shape of the parabola (upwards for $a > 0$, downwards for $a < 0$) and its x-intercepts will show where the graph is above or below the x-axis, directly yielding the solution. Remember to use strict inequality symbols for ranges where the graph does not include the x-intercepts, and non-strict symbols if it does.

Solving Rational Inequalities (Algebraic Fractions)

When inequalities involve algebraic fractions, such as $\frac{P(x)}{Q(x)} > 0$, it's generally best to avoid cross-multiplication if the sign of the denominator is unknown, as this can lead to errors. Instead, move all terms to one side to get zero on the other side. Combine the terms into a single fraction. For example, solve $\frac{2}{x+1} < 1$ by rewriting it as $\frac{2}{x+1} - 1 < 0$, which simplifies to $\frac{2-(x+1)}{x+1} < 0$, or $\frac{1-x}{x+1} < 0$. Then, identify the values of $x$ that make the numerator zero ($x=1$) and the denominator zero ($x=-1$). These critical values divide the number line into intervals. Test a value from each interval to see if the inequality holds. You need to consider the signs of both the numerator and the denominator. For $\frac{\text{Numerator}}{\text{Denominator}} < 0$, you need (Numerator positive AND Denominator negative) OR (Numerator negative AND Denominator positive). Always remember that the denominator cannot be zero.

Representing Solutions

Solutions to inequalities are typically represented using inequality notation (e.g., $x < 5$), interval notation (e.g., $(-\infty, 5)$), or on a number line. On a number line, use an open circle ($\circ$) for strict inequalities ($<$, $>$), indicating that the endpoint is not included in the solution. Use a closed circle ($\bullet$) for non-strict inequalities ($\le$, $\ge$), indicating that the endpoint IS included. Shade the region of the number line that satisfies the inequality. For compound inequalities like $-2 < x ext{ extle } 3$, shade the region between $-2$ and $3$, with an open circle at $-2$ and a closed circle at $3$.

Graphing Basic Linear Inequalities

To graph an inequality like y > mx + c or ax + by ≤ c, first, treat the inequality as an equation (e.g., y = mx + c) and draw the line. This line is your boundary. If the inequality uses < or >, use a dashed (or dotted) line to show that points on the line are NOT part of the solution. If the inequality uses ≤ or ≥, use a solid line to show that points on the line ARE part of the solution. After drawing the line, choose a test point not on the line (the origin (0,0) is often easiest if the line does not pass through it). Substitute the coordinates of the test point into the original inequality. If the inequality is true, shade the region containing the test point. If it's false, shade the region on the other side of the line. This shaded region represents all points (x, y) that satisfy the inequality.

Systems of Inequalities in Two Variables

When solving systems of inequalities, you need to find the region where all inequalities are simultaneously true. Graph each inequality separately on the same coordinate plane, following the rules for solid/dashed lines and shading for each. The solution to the system is the region where all the shaded areas overlap. This overlapping region must be clearly indicated, for example, by shading it darker or labelling it 'R' for region. Any point within this overlapping region (but not on a dashed boundary) will satisfy all inequalities in the system.

Recognising Solution Regions

It's important to be able to identify the solution region from a given set of inequalities or to write the inequalities that describe a given region. Common inequalities associated with axes are x ≥ 0 (region to the right of the y-axis), x ≤ 0 (region to the left of the y-axis), y ≥ 0 (region above the x-axis), and y ≤ 0 (region below the x-axis). For inequalities like y > x, the region is above the line y=x. For y < x, it's below. For x + y < k, it's below the line x + y = k. Always check with a test point. Remember that lines with positive gradients (m > 0) generally have 'above' for > and 'below' for <. Lines with negative gradients (m < 0) can be tricky; sometimes 'above' means less than in terms of y-intercepts. A test point is always the most reliable method.

Practical Application and Higher Reasoning

Higher-tier questions often involve more complex boundary lines (e.g., quadratic or circular boundaries, though the focus here is linear), multiple inequalities, or require you to interpret a solution region in context. You might be asked to find the integer coordinates within a region, or to determine the maximum/minimum value of an expression over a graphically defined region (this often involves linear programming concepts at a higher level, but for GCSE it typically means finding vertices of the feasible region). The ability to accurately draw lines, determine solid/dashed status, correctly shade, and identify the correct overlapping region is fundamental for these more advanced problems. Always label your lines and clearly indicate your final solution region.