1.6 Solving Linear Inequalities

In this lesson you will: Solve simple inequalities. Solve compound inequalities.

These are linear inequalities in one variable: X<1 2n – 3> 9 A Solution of an inequality in one variable is a value of the variable that makes the inequality true.

Consider this true Inequality 1<5 Add 2 to both sides of the inequality. Is the new inequality still true? Subtract 2, Multiply 2 and Divide each side of the inequality by 2. In each case, is the new inequality still true? What would happen if you multiplied or divided both sides of the inequality by -2?

Here are some general conclusions about the operations you can perform on a true inequality to produce another true inequality: Add (or subtract) the same number to both sides. Multiply (or divide) both sides by the same positive number. Multiply (or divide) both sides by the same negative number and reverse the direction of the inequality.

Graphs can help visualize the solutions of an inequality. Can you match the inequality with it’s graph? What is the difference between “closed circles” and “open circles”?

Solve and Graph: 5y - 8 < 12 Add 8 to both sides 5y < 20 Divide both sides by 5 y < 4 Graph the solution

Subtract 6x from both sides Solve and Graph: 2x + 1 < 6x - 1 Subtract 6x from both sides -4x+1 < -1 Subtract 1 from both sides -4x < -2 Divide both sides by -4 and reverse the inequality x > ½ Graph the solution

-2 < x < 1 x < -1 or x > 2 A compound inequality is two simple inequalities joined by “and” or “or”. -2 < x < 1 All real number that are greater than or equal to -2 and less than 1 x < -1 or x > 2 All real number that are less than -1 or greater than or equal to 2

Solving an “And” Compound Inequality Solve – 2 < 3t – 8 < 4 Add 8 to each “part”. +8 +8 +8 6 < 3t < 12 Divide each “part” by 3. 3 3 3 2 < t < 4 Graph the solution All real numbers greater than or equal to 2 AND less than or equal to 4

Solving an “OR” Compound Inequality Separate the inequality into 2 simple parts Solve 2x + 3 < 5 or 4x – 7 > 9 Solve each part separately. 2x + 3 < 5 4x – 7 > 9 x < 1 x > 4 All real numbers less than 1 OR greater than 4

Homework 13-47 odd, 52, 53