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Linear Inequalities

Linear Inequalities. Objective: To graph and solve linear inequalities. Linear Inequality. An inequality is an open mathematical sentence formed when an inequality symbol is placed between two expressions. Examples of inequalities : x < 3 x – 4 ≤ 5 x + 2 > 7 x ≥ 4.

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Linear Inequalities

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  1. Linear Inequalities Objective: To graph and solve linear inequalities

  2. Linear Inequality An inequality is an open mathematical sentence formed when an inequality symbol is placed between two expressions. Examples of inequalities: x < 3 x – 4 ≤5 x + 2 > 7 x ≥ 4 Each of these symbols represent a mathematical directive

  3. Opens away from variable Less Than < Less than or Equal Too Opens away from variable with a line under it ≤ > Opens towards Variable Greater Than Greater than or Equal too Opens towards variable with a line under it ≥ Lets look at the previous slide. Tell me what each statement represents.

  4. The graph of a linear inequality is the graph on the real number line of all solutions of the inequality. Notice the open dot: used when we use < or > Lets graph a simple inequality: x < 3 -4 -2 0 2 4 6 8 In words this says: all real numbers less than 3 Lets graph another: Notice the solid dot: used when we use  or  x ≥ 4 -4 -2 0 2 4 6 8 In words: all real numbers greater than or equal to 4

  5. Solving Linear inequalities Solving an inequality is just like solving a linear equation with one important difference. When you multiply or divide both sides of an inequality by a negative number you must REVERSE the sign

  6. Example 1: Solving a linear inequality (one transformation) Solve x + 5 ≥ 3 Solution: First subtract 5 from both sides x + 5 ≥ 3 - 5 -5 x -2 ≥ Check your answer by graphing on a number line

  7. Example 2: Using more than one transformation Solve 2y – 5 < 7 Solution: 2y – 5 < 7 Add 5 to both sides +5 +5 2y < 12 Now divide both sides by 2 2 2 y < 6 The solution is all numbers less than 6 Check by substituting numbers less than 6 to see if correct

  8. Example 3: Reversing an inequality Solve 5 – 2x > 4 Solution: Subtract 5 from both sides 5 – 2x > 4 -5 -5 Divide both sides by -2 -2x > -1 -2 -2 We divided by a negative number so reverse the sign 1 x < 2 Notice the sign changed from greater than to less than

  9. Example 4: Solving inequalities with variables on both sides Solve 2x – 4 ≥ 4x – 1 Add 4 to both sides 2x ≥ 4x + 3 Subtract 4x from both sides to get the x’s on one side -2x ≥ 3 x ≤ -3/2 Divide both sides by a -2 Notice the sign changed

  10. Review • Write the linear inequality for all real numbers greater than or equal to 4. • Write the verbal phrase that describes x ≤ -2. • Is the number 4.78 a solution of x < 5? Is the number 6 a solution of x< 6? • Solve the inequality 5 – 2x ≥ 3. Then graph it.

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