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Vocabulary and Signs Linear Inequalities Objective: To graph intervals and solve inequalities. Vocabulary and Signs z Linear inequality – Ax + B < C z Solution of an inequality- any number that makes the inequality true. g z Signs z < Less than < Less than or equal to z > Greater than > Greater than or equal to z Graph - a number line indicating solutions of any inequality. Interval Notation z Linear inequality: Ax + B < C z Solution of an inequality- any number that makes the inequality true. g z Signs z < Less than < Less than or equal z > Greater than > Greater than or equal z Graph: a number line indicating solutions of any inequality. zInterval notation: used to represent solution sets zopen interval: use parenthesis ( ); use with < or >; does not include endpoint zclosed interval: use brackets [ ]; use with < or > ; includes endpoint zNote: An interval can be half-open. Graph each inequality. Write using interval notation. Interval Notation Infinite intervals z (a,∞) represents the set of real numbers greater than a; {x | x > a} ( (-----------------a z (-∞,b] represents the set of real numbers less than or equal to b; {x | x < b} -------------------] b z (-∞,∞) represents all real numbers. z A) x < -1 B) x > -3 -------------------) -1 [------------------3 (−∞ ∞, −1) [3 ∞) [3, z C) -4[------------------) <x<2 -4 2 (−4, 2) 1 Addition Property of Inequalities… are the same as with equations… Use the GOLDEN RULE RULE. Examples: Solve, check, and graph. Write using interval notation. zd) p + 6 < 8 p<2 (−∞ ∞, 2) zNote: Always write an inequality with the variable on the left. -------------------) 2 e) 8x < 7x – 6 x < -6 (−∞, −6) -------------------) -6 Multiplication Property Solving with Multiplication z Same as with equations with one important difference: If you multiply or divide by a negative number number, you must REVERSE the inequality symbol. Multiply both sides of each inequality by -5. f) 7 < 8 g) -1 > -4 -35 < -40 5 > 20 Both of these statements are false. This is why we MUST reverse the inequality sign when multiplying or dividing by a negative. Examples: Solve, check and graph. Write using interval notation. Examples: Solve, check and graph. Write using interval notation. −m j) < −3 9 z h) 2x < -10 x < -5 (−∞, −5)) -------------------) -5 i) -7k > 8 k≥ −8 7 −8 ⎞ ⎛ ∞, ⎟ ⎜ −∞ 7 ⎠ ⎝ -2 [-----------------1 0 Multiply both sides by -9. m > 27 (-----------------------------27 2 Examples: Solve, graph, and check. Write using interval notation. STEPS 1. Simplify. (Distribute, combine like terms, eliminate fractions,…) 2 Get variable on one side by itself by 2. performing the inverse operation. 3. Graph the solution on a number line. 4. Verify your solution. Examples: Solve, graph, and check. Write using interval notation. z l) 5 – 3(m – 1) < 2(m + 3) + 1 5 – 3m + 3 < 2m + 6 + 1 -3m + 8 < 2m + 7 -5m < 1 1 ⎡ 1 ⎤ m≥− ⎢⎣ − 5 , ∞ ⎥⎦ 5 -1 [----------------------0 z k) 2k – 5 > 1 + k k–5>1 k>6 [-----------------------------6 Examples: Solve, graph, and check. 1 3 m) (m + 3) + 2 ≤ (m + 8) 4 4 Multiply everything by LCD = 4. ⎡1 ⎤ ⎡3 ⎤ 4 ⎢ ( m + 3) ⎥ + 4(2) ≤ 4 ⎢ ( m + 8) ⎥ ⎣4 ⎦ ⎣4 ⎦ 1(m + 3) + 8 ≤ 3(m + 8) m + 3 + 8 ≤ 3m + 24 −2m + 11 ≤ 24 −2m ≤ 13 m≥− 13 2 ⎡ −13 ⎤ ⎢⎣ 2 , ∞ ⎥⎦ -7 [-----------------------6 Summary z A solution of an inequality is any number that makes the inequality true. z Graphs G h off inequalities i liti iindicate di t allll possible ibl solutions. 3