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! 3A Example Solve the following Simultaneous Equations by Elimination 1 2 x3 Add Substitute x in to ‘2’ 2
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! 3A Example Solve the following Simultaneous Equations by Elimination 1 2 x2 Subtract Substitute y in to ‘2’ 2 x3
Equations and Inequalities 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. 3B Example Solve the following Simultaneous Equations by Substitution 1 2 Rearrange Replace the ‘y’ in equation 2, with ‘2x – 1’ 2 Replace y Expand Sub into 1 or 2
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 3C Example Solve the following Simultaneous Equations by Substitution 1 2 Re-arrange Replace the ‘x’ in equation 2, with ‘3 – 2y’ 2 Expand Brackets Simplify Multiply by -1 Factorise Solve Sub each value for y into one of the equations y = - 1 / 2 y = -1 x = 4, y = - 1 / 2 x = 5, y = -1
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 3C Example Solve the following Simultaneous Equations by Substitution 1 2 Re-arrange Replace the ‘y’ in equation 2, with ‘ 3x – 1 / 2 ’ 2 Replace y Square top and bottom separately Multiply each part by 4 Group on one side or Factorise Solve
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 3C Example Solve the following Simultaneous Equations by Substitution 1 2 Re-arrange or x = -33 / 13 x = 3 x = -33 / 13, y = -56 / 13 x = 3, y = 4
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 > < 3D Example Find the set of values of x for which: Add 5 Divide by 2
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 > < 3D Example Find the set of values of x for which: Subtract x Subtract 9 Divide by 4
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 > < 3D Example Find the set of values of x for which: Subtract 12 Divide by 3 Multiply by -1 REVERSES THE SIGN
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 > < 3D Example Find the set of values of x for which: Expand brackets (careful with negatives) Add 2x and group Add 15 Divide by 5
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 > < 3D Example Find the set of values of x for which: Subtract x and Add 5 Divide by 2 Subtract x Divide by x < 6.5 x > -2
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 > < 3D Example Find the set of values of x for which: Add x and Add 5 Divide by 2 Add 3x Minus 5 Divide by x > 3 x < 2 No answers that work for both…
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 > < 3D Example Find the set of values of x for which: Subtract 7 and Divide by 4 Subtract 11 Divide by x > -1 x > 3
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 x- axis 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 3E Example Find the set of values of x for which: Factorise y x 5 We want values below 0
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 x- axis 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 3E Example Find the set of values of x for which: Factorise y x 5 We want values above 0 Separate sections mean separate inequalities
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 x- axis The graph will be n-shaped, looking at the original equation Then think about which area satisfies the original inequality 3E Example Find the set of values of x for which: Multiply by -1 y x We want values below 0 Factorise
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