Section 4 Solving Inequalities Chapter 1 Section 4 Solving Inequalities

Inequalities The solutions include more than one number Ex: 2 < x ;values that x could be include 3, 7, 45… All of the rules for solving equations apply to inequalities, with one added: If you multiply or divide by a NEGATIVE you must FLIP the sign. (< becomes > and > becomes <) When graphing on a number line: Open dot for < or > Closed (solid) dot for ≤ or ≥ The shading should be easy to see (a slightly elevated line is ok) --- see examples

Solving Inequalities –2x < 3(x – 5) ALGEBRA 2 LESSON 1-4 Solve –2x < 3(x – 5). Graph the solution. –2x < 3(x – 5) –2x < 3x – 15 Distributive Property –5x < –15 Subtract 3x from both sides. x > 3 Divide each side by –5 and reverse the inequality.

Inequalities – Special Solution Cases If the variables cancel, and you’re left with a true statement (ex. 0<10) then all numbers are solutions for the inequality. If the variables cancel, and you’re left with a false statement (ex. 0>10) then no numbers are solutions for the inequality.

Solve 7x ≥ 7(2 + x). Graph the solution. 7x ≥ 14 + 7x Distributive Property 0 ≥ 14 Subtract 7x from both sides. The last inequality is always false, so 7x ≥ 7(2 + x) is always false. It has no solution.

Compound Inequalities - Disjuctions Compound Inequality – a pair of inequalities joined by or For or statements the value must satisfy one of the inequalities Example: x < -1 or x ≥ 3

Or Inequalities ALGEBRA 2 LESSON 1-4 Graph the solution of 3x + 9 < –3 or –2x + 1 < 5. 3x + 9 < –3 or –2x + 1 < 5 3x < –12 or -2x < 4 x < –4 or x > –2

Try This Problem Graph the solution of x – 1 < 3 or x - 3 > 8

Compound Inequalities - Conjuctions A pair of inequalities joined by an and statement The value must satisfy both inequalities. In other words, the solution is where the inequalities overlap. Can be written in two forms: Example: -1 < x and x ≤ 3 is the same as -1 < x ≤ 3

And Inequalities Graph the solution of 3x – 1 > -28 and 2x + 7 < 19. 3x > -27 and 2x < 12 x > -9 and x < 6 Graph the solution of -8 < 3x + 1 <19 -9 < 3x < 18 -3 < x < 6

Check for Understanding Graph the solution of 2x > x + 6 and x – 7 < 2 x > 6 and x < 9