0.6 Solving Absolute Value Inequality 9/7/2012

Interval Notation. Interval notation translates the information from the real number line into symbols. For example becomes the interval (-2,5]. To indicate that an endpoint is included, we use a square bracket; to exclude an endpoint, we use parentheses. Example is written in interval notation as (- , 3]. The infinity symbols “  ” and " -  " are used to indicate that the set is unbounded in the positive (  ) or negative (-  ) direction of the real number line. "  " and " -  " are not real numbers, just symbols. Therefore we always exclude them as endpoints by using parentheses. If the set consists of several disconnected pieces, we use the symbol for union "  ":

GivenSolutionGraph When C is positive l x l < C - C < x < C l x l ≤ C - C ≤ x ≤ C l x l > C x > C or x < -C l x l ≥ C x ≥ C or x ≤ - C Remember: > “GreatOR than” or ≥ “GreatOR than or equal to” is an OR problem

GivenSolution When C is negative l x l < -C No solution l x l ≤ -C No solution l x l > -C All real numbers l x l ≥ - C All real numbers

GivenSolution When C is Zero l x l < 0 No solution l x l ≤ 0 x = 0 l x l > 0 x > 0 or x < 0 l x l ≥ 0 All real numbers

Solve |x| < 3. Graph the solution and write solution in interval notation. Answer: -3 < x < 3 Interval Notation: (-3, 3)

Solve |x| ≥ 4. Graph the solution and write solution in interval notation. Answer: x ≤ -4 or x ≥ 4 Interval Notation: (- , -4]  [4, +  )

Example 1 Solve an Inequality of the Form Solve. Then graph the solution. Write solution in interval notation. + xb ≤ c + x4 ≤ 10 SOLUTION + x4 ≤ 10 Write original inequality. – + x4 ≤ 10 Write equivalent double inequality. 10 ≤ – x ≤ 6 Subtract 4 from each expression. 14 ≤ 16 –. 14 – 12 – 10 – 8 – 6 – 4 – 2 – Interval Notation: [-14, 6]

Example 2 Solve an Inequality of the Form Solve. Then graph the solution. Write solution in interval notation. + axbc < + 2x2x37 < SOLUTION Write original inequality. + 2x2x37 < – Write equivalent compound inequality. + 2x2x37 < 7 < – Subtract 3 from each expression. 10 2x2x4 << – Divide each expression by 2. 5 x2 << 6 – 5 – 4 – 3 – 0122 – 1 – 3 Interval Notation: (-5, 2)

Example 3 Solve an Inequality of the Form Solve. Then graph the solution. + axbc≥ x –≥ FIRST INEQUALITYSECOND INEQUALITY x – ≤ – x ≥– Add 1 to each side. x2 2 1 ≤ – 4x 2 1 ≥ 6 – 4 – 022 – Interval Notation: (- , -4]  [8,  )

Example 4 Solve an Inequality of the Form Solve. Then graph the solution. + axb-c≤ 3x1-5 –≤ From the table, the inequality has no solution

Example 5 Solve an Inequality of the Form Solve.. + axb-c≥ 4x2-6 –≥ From the table, the solution is all real numbers.

Example 6 Solve an Inequality of the Form Solve +. + axb0< x30 –< From the table, the inequality has no solution

Example 7 Solve an Inequality of the Form Solve +. + axb0≤ 2x40 –≤ 2x + 4 = x = x = -2

Example 7 Solve an Inequality of the Form Solve. Then graph the solution. + axb0≥ 2x10 +> 2x + 1 > x > x > -½ 2x + 1 < x > x < -½ 6 – 4 – 022 – Interval Notation: (- , -½)  (-½,  )

Homework: WS 0.6 #1-18, 27, 29