Solve Exponential and Logarithmic Inequalities

Vocabulary  An exponential inequality in one variable is an inequality that can be written in the form ab x + k 0, or ab x + k > 0, where a = 0, b > 0, and b = 1.  A logarithmic inequality in one variable is an inequality that can be written in the form log b x + k 0, or log b x + k > 0, where b > 0, and b = 1.

Solving Exponential and Logarithmic Inequalities… YYou can use the same methods used to solve exponential and logarithmic inequalities as we learned to solve equations.

Example 1:  Solve 4 x + 1 > 32 Step 1: Write the equation that corresponds to inequality. Step 2: Solve for x. Step 3: plot the x-value on a number line and test intervals [1.5, ∞)

You Try:  Solve the exponential inequality x – 1 < x – 4 < 18

Example 2:  Solve the logarithmic inequality 1. log 5 x < 2  Step 1: Write the corresponding equation  Step 2: solve for x.  Step 3: plot the x-value on a number line and test intervals (0, 25) WATCH OUT! How many x-values should you plot on the number line?

Example 2 Cont’d:  Solve the logarithmic inequality 2. log 4 x + 8 > 11  Step 1: Write the corresponding equation  Step 2: solve for x.  Step 3: plot the x-value on a number line and test intervals [64, ∞)

You Try:  Solve the logarithmic inequalities 1. log 6 2x + 7 < log 9 (x – 8) > 3/2

Homework:  P.