Equations and Inequalities

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Introduction to Equations and Inequalities

An equation is a mathematical statement showing that two expressions are equal, while an inequality shows that one expression is greater than, less than, or not equal to another. Solving equations and inequalities means finding the value or range of values of the unknown variable that make the statement true, a skill that is central to algebra and widely applied throughout mathematics and science.

Solving Linear Equations

A linear equation is one in which the unknown variable appears only to the power of one, such as 2x + 3 = 11. Linear equations are solved by performing the same operation to both sides of the equation in order to isolate the variable, while keeping the equation balanced. For the example above, subtracting 3 from both sides gives 2x = 8, and dividing both sides by 2 gives x = 4, which can be checked by substituting back into the original equation.

Equations with Unknowns on Both Sides

Some equations have the unknown variable appearing on both sides, such as 5x + 2 = 2x + 14. These are solved by first collecting all the variable terms on one side and all the constant terms on the other, for example subtracting 2x from both sides gives 3x + 2 = 14, and subtracting 2 from both sides gives 3x = 12, leading to x = 4 after dividing by 3.

Equations Involving Brackets and Fractions

When an equation contains brackets, these should generally be expanded first before collecting like terms and solving as normal. When an equation contains fractions, it is often easier to eliminate the fractions first by multiplying every term in the equation by the lowest common denominator, transforming the equation into a simpler form without fractions before continuing to solve for the unknown.

Solving Quadratic Equations by Factorising

A quadratic equation contains a term where the unknown variable is raised to the power of two, such as x² - 5x + 6 = 0. One method of solving a quadratic equation is factorising it into two brackets, such as (x - 2)(x - 3) = 0, and then using the fact that if two factors multiply to give zero, at least one of them must itself be zero, giving the two solutions x = 2 and x = 3.

The Quadratic Formula

Not every quadratic equation factorises easily, so the quadratic formula provides a reliable method for solving any quadratic equation of the form ax² + bx + c = 0. The formula states that x equals negative b, plus or minus the square root of b squared minus 4ac, all divided by 2a. Substituting the values of a, b, and c from a specific quadratic equation into this formula allows both solutions to be calculated directly, even when the equation cannot be factorised using whole numbers.

Simultaneous Equations

Simultaneous equations involve two or more equations that must be solved together to find values of the unknowns that satisfy all the equations at once. Two common methods for solving simultaneous linear equations are substitution, where one equation is rearranged to express one variable in terms of the other and then substituted into the second equation, and elimination, where the equations are added or subtracted to eliminate one of the variables entirely.

Understanding Inequalities

An inequality compares two expressions using symbols such as greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤). Solving a linear inequality follows the same process as solving a linear equation, with one important exception: when multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign must be reversed.

Representing Inequalities on a Number Line

Solutions to inequalities are often shown on a number line, using an open circle to indicate a value that is not included (used for strict inequalities such as > or <) and a closed, filled-in circle to indicate a value that is included (used for ≥ or ≤). An arrow is then drawn from the circle in the direction of all the values that satisfy the inequality, providing a clear visual representation of the full range of solutions.

Worked Example: Solving Simultaneous Equations by Elimination

Consider the equations 2x + y = 11 and x - y = 1. Adding the two equations together eliminates y, since +y and -y cancel, giving 3x = 12, so x = 4. Substituting x = 4 back into the second equation gives 4 - y = 1, so y = 3. These values can be checked by substituting both x = 4 and y = 3 into the original first equation, confirming that 2(4) + 3 = 11, which is correct.

Forming Equations from Real-World Problems

Equations are often used to model real-world situations, such as calculating the number of items that can be bought within a budget or finding an unknown age based on given clues. Translating a word problem into an equation requires carefully identifying the unknown quantity, assigning it a variable, and expressing the relationships described in the problem using that variable, before solving the resulting equation using the appropriate method.

Solving Quadratic Inequalities

A quadratic inequality, such as x² - 5x + 6 > 0, is solved by first finding the critical values where the corresponding quadratic equation equals zero, in this case x = 2 and x = 3, by factorising or using the quadratic formula. These critical values divide the number line into regions, and testing a value from each region in the original inequality reveals which regions satisfy the inequality, allowing the full solution set to be stated clearly, often as a combination of two separate inequalities.

Checking Solutions

Whatever method is used to solve an equation, it is good practice to check the solution by substituting it back into the original equation to confirm both sides are equal. For simultaneous equations, the solution should be checked in both original equations, not just one, since a value might satisfy one equation by coincidence while still being an incorrect solution to the system as a whole. This checking step helps catch arithmetic slips made during the solving process before an answer is finalised.

Key Terms to Remember

• Linear equation: an equation in which the unknown appears only to the power of one.
• Quadratic equation: an equation containing a term where the unknown is raised to the power of two.
• Simultaneous equations: two or more equations solved together to find values satisfying all of them.
• Inequality: a statement comparing two expressions using greater than, less than, or similar symbols.

Summary

Equations and inequalities allow unknown quantities to be found or constrained based on given mathematical relationships. From simple linear equations to quadratic equations solved by factorising or the quadratic formula, and from simultaneous equations to inequalities represented on a number line, these techniques form some of the most widely applied skills across the entire mathematics syllabus.