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Linear Equations & Inequalities (Cambridge (CIE) IGCSE Maths)

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  • FrontSolving Linear Equations

    What is the greatest power of x in a linear equation?

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  • What is the greatest power of x in a linear equation?

    The greatest power of x in a linear equation is 1.

    There is no term in x2 or any higher power.

  • What does 'isolate the variable' mean?

    Isolate the variable means to get the variable term, e.g. x, by itself on one side of the equation.

  • What are inverse operations?

    Inverse operations are the opposite operations to what has already happened to the variable.

    E.g. In the expression 4 x, x is being multiplied by 4.
    The inverse operation is to divide by 4.

    Examples of inverse operations are:

    • Add and subtract

    • Multiply and divide

    • Square and take the square root

  • True or False?

    To solve a linear equation, e.g. 3 x plus 4 equals 13, you need to isolate the variable by carrying out inverse operations to both sides.

    True.

    To solve a linear equation, you need to isolate the variable, (get the letter on its own), by carrying out inverse operations to both sides.

    E.g.

    3 x plus 4 equals 13 table row cell negative 4 end cell cell space space space space space space space space space space space space space space space space space space space space minus 4 end cell end table 3 x equals 9 divided by 3 space space space space space space space space space space space space space space space space space space space space space divided by 3 x equals 3

  • What is the order of inverse operations to solve a linear equation of the form a x plus b equals c to find x.

    To solve a linear equation of the form a x plus b equals c:

    • First subtract b from both sides

    • Then divide both sides by a

  • True or False?

    If a linear equation contains the unknown variable on both sides, you should first collect the variable terms on one side.

    E.g. 8 x plus 7 equals 2 x plus 11.

    True.

    If a linear equation contains the unknown variable on both sides, you should first collect the variable terms on one side.

    E.g.

    table row cell 8 x plus 7 end cell equals cell 2 x plus 11 end cell row cell 6 x plus 7 end cell equals 11 end table

  • What is the first step to solve a linear equation with brackets?

    E.g. 4 open parentheses x plus 6 close parentheses equals 32.

    The first step to solve a linear equation with brackets is to expand the brackets (on both sides if necessary).

    E.g.

    table row cell 4 open parentheses x plus 6 close parentheses end cell equals 32 row cell 4 x plus 24 end cell equals 32 end table

  • What is the first step to solve a linear equation with fractions?

    E.g. fraction numerator x plus 9 over denominator 2 end fraction equals fraction numerator 3 x minus 1 over denominator 4 end fraction.

    The first step to solve a linear equation with fractions is to remove the fractions by multiplying both sides by the denominators.

    E.g.

    table row cell fraction numerator x plus 9 over denominator 2 end fraction end cell equals cell fraction numerator 3 x minus 1 over denominator 4 end fraction end cell row cell 4 open parentheses x plus 9 close parentheses end cell equals cell 2 open parentheses 3 x minus 1 close parentheses end cell row blank blank blank end table

  • After solving a linear equation, how can you check your answer to see if it is correct?

    After solving a linear equation, you should substitute your answer back into the equation to check if it is correct.

    This is a quick way to spot if you have made any mistakes.

  • Define the word inequality in algebra.

    An inequality compares a left-hand side to a right-hand side and states which one is bigger, using the symbols less than comma space greater than comma space less or equal than comma space greater or equal than.

  • Explain the meaning of the word linear in linear inequality.

    The word linear in linear inequality means that the terms in the inequality are either constant numbers or terms in x, but not terms in x squared or x cubed etc.

    These are examples of linear inequalities:

    • x plus 2 greater than 5

    • 2 x less than x minus 1

  • True or False?

    You can add or subtract terms to both sides of a linear inequality in exactly the same way as you do to a linear equation.

    True.

    You can add or subtract terms to both sides of a linear inequality in exactly the same way as you do to a linear equation.

  • True or False?

    You can multiply or divide both sides of a linear inequality in exactly the same way as you do to a linear equation.

    False.

    You can multiply or divide both sides of a linear inequality in exactly the same way as you do to a linear equation as long as you multiply or divide by positive numbers.

    If, however, you multiply or divide both sides by negative numbers, you have to flip the direction of the inequality sign.

    E.g. You can divide table row cell 2 x end cell less than 4 end table by 2 to get table row x less than 2 end table.
    You can divide table row cell negative 2 x end cell less than 4 end table by -2, but you must flip the inequality to get table row x greater than cell negative 2 end cell end table.

  • How do number lines highlight the difference between strict inequalities (such as x less than 3) and non-strict inequalities (such as x less or equal than 3)?

    Number lines show an open circle for strict inequalities, e.g. x less than 3, and a closed circle for non-strict inequalities, e.g. x less or equal than 3.

    Two number lines labelled from -5 to +5. The top number line shows an arrow from 3, marked by an open circle, to negative direction. It is labelled 'x < 3'. The bottom number line shows an arrow from 3, marked by a closed circle, to negative direction. It is labelled 'x ≤ 3'.

  • True or False?

    The number line representing " x less than 1 or x greater than 3" consists of two separate arrows pointing outwards in opposite directions.

    True.

    The number line representing " x less than 1 or x greater than 3" consists of two separate arrows pointing outwards in opposite directions.

    A number line from -5 to 5. An empty circle is at 1 with an arrow pointing in the negative direction. Another empty circle is at 3 with an arrow pointing in the positive direction.

  • True or False?

    The diagram below shows the inequality negative 4 less than x less or equal than 2.

    A number line from -5 to 5. Empty circles are located at -4 and 2 with a horizontal line joining them.

    False.

    The number line in the diagram does not show the inequality negative 4 less than x less or equal than 2.

    It has two open circles which indicate the inequality negative 4 less than x less than 2.

  • Explain how you would solve an inequality in the form negative 10 less than a x plus b less than 10.

    To solve an inequality in the form negative 10 less than a x plus b less than 10,

    1. Subtract b from all three parts.

    2. Then divide all three parts by a.

    E.g. negative 10 less than 2 x plus 6 less than 10 minus 16 less than 2 x less than 16 minus 8 less than x less than 8

    An alternative method is to split into two different inequalities, negative 10 less than 2 x plus 6 and 2 x plus 6 less than 10, then solve these individually.

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