Solving Equations And Inequalities
Overview
An equation is a question about a number: which value makes these two expressions agree? Solving it is the process of stripping away everything that has been done to the unknown until it stands alone. That is the mechanical half of this topic. The other half is harder and worth more marks: turning a situation described in words, or drawn as a shape, into the equation in the first place.
You will solve linear equations, including those with brackets and with the unknown on both sides, solve quadratic equations by factorising, solve two linear simultaneous equations in two variables, find approximate solutions from a graph, translate simple situations or procedures into algebraic expressions or formulae, derive and solve equations from geometrical problems and problems set in context, and solve linear inequalities and represent the solution set on a number line. On the Extension Tier you will add completing the square and the quadratic formula, simultaneous equations where one is quadratic, inequalities in two variables and quadratic inequalities, and the graphical representation of a solution set.
Objectives
- [Core] solve linear equations in one unknown algebraically
- [Core] find approximate solutions using a graph (Notes: including use of brackets and those with the unknown on both sides of the equation)
- [Core] solve quadratic equations algebraically by factorising
- [Core] find approximate solutions using a graph
- [Extension] including completing the square and by using the quadratic formula
- [Core] solve two linear simultaneous equations in two variables algebraically
- [Core] find approximate solutions using a graph
- [Extension] including one linear and one quadratic
- [Core] translate simple situations or procedures into algebraic expressions or formulae
- [Core] derive an equation (or two simultaneous equations), solve the equation(s) and interpret the solution (Notes: including the solution of geometrical problems and problems set in context)
- [Core] solve linear inequalities in one variable
- [Core] represent the solution set on a number line
- [Extension] solve linear inequalities in one or two variable(s), and quadratic inequalities in one variable
- [Extension] represent the solution set on a number line and on a graph (Notes: students should know the conventions of an open circle on a number line for a strict inequality and a closed circle for an included boundary. In graphical work the convention of a dashed line for strict inequalities and a solid line for an included inequality will be required)
Mind map
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Lesson Note
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Lesson Evaluation
Congratulations on completing the lesson on Solving Equations And Inequalities. Now that youve explored the key concepts and ideas, its time to put your knowledge to the test. This section offers a variety of practice questions designed to reinforce your understanding and help you gauge your grasp of the material.
You will encounter a mix of question types, including multiple-choice questions, short answer questions, and essay questions. Each question is thoughtfully crafted to assess different aspects of your knowledge and critical thinking skills.
Use this evaluation section as an opportunity to reinforce your understanding of the topic and to identify any areas where you may need additional study. Don't be discouraged by any challenges you encounter; instead, view them as opportunities for growth and improvement.
- Circle the equation with roots 4 and -8. A. 4x(x - 8) = 0 B. (x - 4)(x + 8) = 0 C. x^2 - 32 = 0 D. (x + 4)(x - 8) = 0 Answer: B
- A number line shows a closed circle at -7, an open circle at 6, and the values between them shaded. Which inequality does it show? A. -7 < x < 6 B. -7 is less than or equal to x, and x < 6 C. -7 < x, and x is less than or equal to 6 D. -7 is less than or equal to x, and x is less than or equal to 6 Answer: B
- Solve 5(x + 4) = 3(x + 7) + 2. A. x = 0.5 B. x = 1.5 C. x = 3 D. x = 21.5 Answer: B
- How many solutions does x^2 = 5x have? A. none B. one C. two D. infinitely many Answer: C
- Which method must be used to solve the pair 4x + y = -3 and y = x^2 + 2x + 5? A. adding the two equations B. subtracting the two equations C. substitution D. trial and improvement Answer: C
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Work through these questions in the app
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