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2.7 Absolute Value Inequalities

Linear Inequalities An inequality is a statement that contains one of the symbols:, ≤ or ≥. EquationsInequalities x = 3x > 3 12 = 7 – 3y12 ≤ 7 – 3y

Graphing Solutions A solution of an inequality is a value of the variable that makes the inequality a true statement. The solution set of an inequality is the set of all solutions.

Graphing Solutions

Graph each set on a number line and then write it in interval notation. a. b. c. Example

Solve: x – 2 < 5 Graph the solution set. Example

Solve: Graph the solution set. Example

Solve: Graph the solution set. Example

Solve: Graph the solution set. Example The inequality symbol is reversed since we divided by a negative number.

Step 1: Clear the equation of fractions or decimals by multiplying both sides of the inequality by the appropriate nonzero number. Step 2: Use distributive property to remove grouping symbols such as parentheses. Step 3: Combine like terms on each side of the inequality. Step 4: Use the addition property of inequality to rewrite the inequality as an equivalent inequality with variable terms on one side and numbers on the other side. Step 5: Use the multiplication property of inequality to get the variable alone on one side of the inequality. Solving Linear Inequalities

If a is a positive number, then  X  < a is equivalent to  a < x < a. Absolute Value Inequalities

Solve  x + 4  < 6  6 < x + 4 < 6  6 – 4 < x + 4 – 4 < 6 – 4  10 < x < 2 (  10, 2) Example

Solve  x  3  + 6  7  x  3   1  1  x  3  1 2  x  4 [2, 4] Example

Solve  8x  3  <  2 No solution. An absolute value cannot be less than a negative number, since it can’t be negative. Example

If a is a positive number, then  X  > a is equivalent to X > a or X <  a. Absolute Value Inequalities

Example Solve for x: The sign means “less than or equal to.” The absolute value of any expression will never be less than 0, but it may equal 0. The solution set is {  1}.

Solve  x  + 1 > 2  x  > x 1 3x  9 x  3 (  , )  (  3,  ) Example

Solve  x + 2   0 The solution is all real numbers, since all absolute values are non-negative. Any value for x we substitute into the inequality will give us a true statement. Example

Solve (  ,  15]  [1,  ) Example Graph of solution ( ,  15]  [1,  )