1.6 Solving Inequalities

Solving Inequalities Solving inequalities follows the same procedures as solving equations. There are a few special things to consider with inequalities: We need to look carefully at the inequality sign. We also need to graph the solution set.

Review of Inequality Signs > greater than < less than greater than or equal less than or equal

How to graph the solutions > Graph any number greater than. . . open circle, line to the right < Graph any number less than. . . open circle, line to the left Graph any number greater than or equal to. . . closed circle, line to the right Graph any number less than or equal to. . . closed circle, line to the left

Solve the inequality: -4 -4 x < 3 x + 4 < 7 -4 -4 x < 3 Subtract 4 from each side. Keep the same inequality sign. Graph the solution. Open circle, line to the left. 3

There is one special case. Sometimes you may have to reverse the direction of the inequality sign!! That only happens when you multiply or divide both sides of the inequality by a negative number.

Example: -3y > 18 Solve: -3y + 5 >23 -5 -5 -5 -5 -3y > 18 -3 -3 y < -6 Subtract 5 from each side. Divide each side by negative 3. Reverse the inequality sign. Graph the solution. Open circle, line to the left. -6

Try these: Solve 2x+3>x+5 Solve - c - 11>23 Solve 3(r-2)<2r+4

Next slide Gives you more detailed explanation on solving inequalities

Solving Inequalities To solve an inequality, use the same procedure as solving an equation with one exception. When multiplying or dividing by a negative number, reverse the direction of the inequality sign. -3x < 6 divide both sides by -3 -3x/-3 > 6/-3 x > -2

Solving Inequalities -4x + 2 > 10 -4x > 8 x < -2 To graph the solution set, circle the boundary and shade according to the inequality. Use an open circle for < or > and closed circles for ≤ or ≥. -2 -1

Solving Inequalities 3b - 2(b - 5) < 2(b + 4) 1

Solving Absolute Value Inequalities Solving absolute value inequalities is a combination of solving absolute value equations and inequalities. Rewrite the absolute value inequality. For the first equation, all you have to do is drop the absolute value bars. For the second equation, you have to negate the right side of the inequality and reverse the inequality sign.

Solve: |2x + 4| > 12 2x + 4 > 12 or 2x + 4 < -12 4 -8

Solve: 2|4 - x| < 10 |4 - x| < 5 -1 < x < 9 4 - x < 5 and 4 - x > -5 - x < 1 - x > -9 x > -1 and x < 9 -1 < x < 9 9 -1