Introduction This Chapter focuses on solving Equations and

Equations and Inequalities Simultaneous Equations You need to be able to solve Simultaneous Equations

Equations and Inequalities Example Solve the following Simultaneous Equations by Substitution Simultaneous Equations You

Equations and Inequalities Simultaneous Equations You will need the substitution method when one of

Equations and Inequalities Solving Inequalities You need to be able to solve Linear Inequalities,

Equations and Inequalities Solving Inequalities Example Find the set of values of x for

Equations and Inequalities Quadratic Inequalities To solve a Quadratic Inequality, you need to: 1)

Summary • We have looked at solving Simultaneous Equations, including Quadratics • We have

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Introduction • This Chapter focuses on solving Equations and Inequalities • It will also make use of the work we have done so far on Quadratic Functions and graphs

Equations and Inequalities Simultaneous Equations You need to be able to solve Simultaneous Equations by Elimination. Remember that the 2 equations can also be drawn on a graph, and the solutions are where they cross. However, this method is less accurate when we start needing decimal/fractional answers. GENERAL RULE If what you’re cancelling have different signs, add. If they have the SAME sign, SUBTRACT! Example Solve the following Simultaneous Equations by Elimination 1 2 x 3 Add Substitute x in to ‘ 2’ 2 3 A

Equations and Inequalities Example Solve the following Simultaneous Equations by Substitution Simultaneous Equations You need to be able to solve Simultaneous Equations by Substitution. This involves using one equation to write y ‘in terms of x’ or vice versa. This is then substituted into the other equation. 1 2 Rearrange Replace the ‘y’ in equation 2, with ‘ 2 x – 1’ 2 Replace y Expand Sub into 1 or 2 3 B

Equations and Inequalities Simultaneous Equations You will need the substitution method when one of the equations is quadratic. Example Solve the following Simultaneous Equations by Substitution 1 You will end up with 0, 1 or 2 answers as with any Quadratic. 2 This means you will either get 0, 1 or 2 pairs of answers 2 Sub each value for y into one of the equations y = -1/2 y = -1 Re-arrange Replace the ‘x’ in equation 2, with ‘ 3 – 2 y’ Expand Brackets Simplify Multiply by -1 Factorise x = 4, y = -1/ 2 x = 5, y = -1 Solve 3 C

Equations and Inequalities Simultaneous Equations You will need the substitution method when one of the equations is quadratic. You will end up with 0, 1 or 2 answers as with any Quadratic. This means you will either get 0, 1 or 2 pairs of answers Factorise or Solve Example Solve the following Simultaneous Equations by Substitution 1 2 Re-arrange Replace the ‘y’ in equation 2, with ‘ 3 x – 1/2’ 2 Replace y Square top and bottom separately Multiply each part by 4 Group on one side 3 C

Equations and Inequalities Simultaneous Equations You will need the substitution method when one of the equations is quadratic. You will end up with 0, 1 or 2 answers as with any Quadratic. Example Solve the following Simultaneous Equations by Substitution Re-arrange 1 2 x= -33/ 13 x=3 This means you will either get 0, 1 or 2 pairs of answers or x = -33/13, y = -56/ 13 x = 3, y = 4 3 C

Equations and Inequalities Solving Inequalities You need to be able to solve Linear Inequalities, sometimes more than one together. An Inequality will give a range of possible answers, rather than specific values (like an Equation would). You can solve them in the same way as a Linear Equation. Example Find the set of values of x for which: Add 5 Divide by 2 Only difference: When you multiply or divide by a negative, you must reverse the sign > < 3 D

Equations and Inequalities Solving Inequalities You need to be able to solve Linear Inequalities, sometimes more than one together. An Inequality will give a range of possible answers, rather than specific values (like an Equation would). You can solve them in the same way as a Linear Equation. Example Find the set of values of x for which: Subtract x Subtract 9 Divide by 4 Only difference: When you multiply or divide by a negative, you must reverse the sign > < 3 D

Equations and Inequalities Solving Inequalities You need to be able to solve Linear Inequalities, sometimes more than one together. An Inequality will give a range of possible answers, rather than specific values (like an Equation would). You can solve them in the same way as a Linear Equation. Only difference: When you multiply or divide by a negative, you must reverse the sign > < Example Find the set of values of x for which: Subtract 12 Divide by 3 Multiply by -1 REVERSES THE SIGN 3 D

Equations and Inequalities Solving Inequalities You need to be able to solve Linear Inequalities, sometimes more than one together. An Inequality will give a range of possible answers, rather than specific values (like an Equation would). You can solve them in the same way as a Linear Equation. Only difference: When you multiply or divide by a negative, you must reverse the sign > < Example Find the set of values of x for which: Expand brackets (careful with negatives) Add 2 x and group Add 15 Divide by 5 3 D

Equations and Inequalities Solving Inequalities Example Find the set of values of x for which: You need to be able to solve Linear Inequalities, sometimes more than one together. An Inequality will give a range of possible answers, rather than specific values (like an Equation would). You can solve them in the same way as a Linear Equation. Only difference: When you multiply or divide by a negative, you must reverse the sign > < and Subtract x Divide by 4 Add 5 Divide by 2 x < 6. 5 -4 -2 0 2 4 6 8 10 x > -2 3 D

Equations and Inequalities Solving Inequalities Example Find the set of values of x for which: You need to be able to solve Linear Inequalities, sometimes more than one together. and An Inequality will give a range of possible answers, rather than specific values (like an Equation would). Add x You can solve them in the same way as a Linear Equation. Divide by 2 Only difference: When you multiply or divide by a negative, you must reverse the sign > < Add 3 x Minus 5 Add 5 Divide by 5 -4 x<2 -2 0 2 4 6 8 10 x>3 No answers that work for both… 3 D

Equations and Inequalities Solving Inequalities Example Find the set of values of x for which: You need to be able to solve Linear Inequalities, sometimes more than one together. An Inequality will give a range of possible answers, rather than specific values (like an Equation would). and Subtract 7 Subtract 11 Divide by 4 Divide by 2 You can solve them in the same way as a Linear Equation. Only difference: When you multiply or divide by a negative, you must reverse the sign > < -4 -2 0 2 4 6 8 10 x > -1 x>3 3 D

Equations and Inequalities Quadratic Inequalities To solve a Quadratic Inequality, you need to: 1) Solve the Quadratic Equation 2) Sketch a graph of the Equation 3) Decide which is the required set of values Remember that the solutions are where the graph crosses the xaxis The graph will be u-shaped. Where it crosses the y-axis does not matter for this topic Then think about which area satisfies the original inequality Example Find the set of values of x for which: Factorise y -1 5 x We want values below 0 3 E

Equations and Inequalities Quadratic Inequalities To solve a Quadratic Inequality, you need to: 1) Solve the Quadratic Equation 2) Sketch a graph of the Equation 3) Decide which is the required set of values Example Find the set of values of x for which: Multiply by 1 Factorise Remember that the solutions are where the graph crosses the xaxis The graph will be n-shaped, looking at the original equation Then think about which area satisfies the original inequality y -3 We want values below 0 0. 5 x 3 E

Summary • We have looked at solving Simultaneous Equations, including Quadratics • We have seen how to solve Inequalities • We have seen how to use graphs to solve Quadratic Inequalities