1 OCF Inequalities MCR3U - Santowski

2 (A) Review (i) Symbols: Inequalities make use of the following symbols: >, >,, >, <, < (meaning either less than, greater than or equal to) (ii) Interpretation: ex. 3x - 1 > 8 is an inequality that reads 3x - 1 is greater than 8 (ii) Interpretation: ex. 3x - 1 > 8 is an inequality that reads 3x - 1 is greater than 8 (iii) Techniques: In working with inequalities, the regular rules for equations hold true => what ever you do to one side of the equation, you must do to the other side of the equation. (iii) Techniques: In working with inequalities, the regular rules for equations hold true => what ever you do to one side of the equation, you must do to the other side of the equation. Only one variation in techniques will be reviewed => when multiplying or dividing through by a negative quantity, reverse the inequality symbol Only one variation in techniques will be reviewed => when multiplying or dividing through by a negative quantity, reverse the inequality symbol

3 (A) Review (iv) Solutions: In solving an inequality, we are looking for every possible value for x that satisfies the given condition. As such, there will be many possible values for the variable (unlike linear or quadratic equations) (iv) Solutions: In solving an inequality, we are looking for every possible value for x that satisfies the given condition. As such, there will be many possible values for the variable (unlike linear or quadratic equations) (v) Notation: Solutions may be presented using set notation, number lines, or interval notation (v) Notation: Solutions may be presented using set notation, number lines, or interval notation (vi) Ex of Set Notation: (vi) Ex of Set Notation: {x | x > -2, x E R}  which becomes [-2, +∞) in interval notation {x | x > -2, x E R}  which becomes [-2, +∞) in interval notation {x | 2 < x < 7, x E R}  which becomes (2,7] in interval notation {x | 2 < x < 7, x E R}  which becomes (2,7] in interval notation {x E R | x < -2}  which becomes (-∞, -2] in interval notation {x E R | x < -2}  which becomes (-∞, -2] in interval notation Ex of Number Lines (Draw on board) Ex of Number Lines (Draw on board)

4 (B) Examples ex 1 Graph each set on a number line ex 1 Graph each set on a number line (a) {x E N | -5 < x < 2} (a) {x E N | -5 < x < 2} (b) {x E R | -5 < x < 2} (b) {x E R | -5 < x < 2} (c) {x E R | x 2 < 16} (c) {x E R | x 2 < 16} (d) {x E R | x 2 > 9} (d) {x E R | x 2 > 9}

5 (B) Examples ex 2 Solve each inequality and present the solution on a number line, in set notation, and in interval notation ex 2 Solve each inequality and present the solution on a number line, in set notation, and in interval notation (a) 1 - x < -3 (a) 1 - x < -3 (b) 2a/3 + 4 > 2 (b) 2a/3 + 4 > 2 (c) -4 < (1 - 3x)/2 < 1 (c) -4 < (1 - 3x)/2 < 1 ex 3 Given a graph of y = 3x - 1. If the values for x (the domain) are between -2 and 7 ie [-2,7], what values are there for y (called the range). ex 3 Given a graph of y = 3x - 1. If the values for x (the domain) are between -2 and 7 ie [-2,7], what values are there for y (called the range).

6 (C) Internet Links (i) atements/LinearInequalities/LinearInequalities.html (i) atements/LinearInequalities/LinearInequalities.html atements/LinearInequalities/LinearInequalities.html atements/LinearInequalities/LinearInequalities.html (ii) (ii) (iii) ab/beg_algebra/beg_alg_tut18_ineq.htm (iii) ab/beg_algebra/beg_alg_tut18_ineq.htm ab/beg_algebra/beg_alg_tut18_ineq.htm ab/beg_algebra/beg_alg_tut18_ineq.htm

7 (D) Homework Nelson text p245, Q1bdghi, 2def, 3acdefh, 4,5bc