Solve absolute value inequalities Section 6.6 #45 The chief forms of beauty are order and symmetry and definiteness, The chief forms of beauty are order and symmetry and definiteness, which the mathematical sciences demonstrate in a special degree. aristotle

Concept Yesterday we looked at how to solve absolute value equalities, today we’re going to look at inequalities Yesterday we looked at how to solve absolute value equalities, today we’re going to look at inequalities This process is very similar with one important difference This process is very similar with one important difference The process also requires us to utilize our understanding of AND and OR statements The process also requires us to utilize our understanding of AND and OR statements

Remember for the other day Solve Solve

Remember for the other day Evaluate Evaluate

Solving with Absolute value Let’s solve this Let’s solve this We let’s analyze this information &

Solving with Absolute value What about this one? What about this one? Let’s analyze this information Or

Solving with Absolute value When we look at this process, we essentially solve the equation twice When we look at this process, we essentially solve the equation twice Once with the positive value and the same sign Once with the positive value and the same sign Once with the negative value and the opposite sign Once with the negative value and the opposite sign This either creates an AND statement or an OR statement This either creates an AND statement or an OR statement &

Solving with Absolute value There isn’t a tried and true rule for which creates an AND statement and which creates an OR statement. It’s typically better to plot the function There isn’t a tried and true rule for which creates an AND statement and which creates an OR statement. It’s typically better to plot the function or

Solving with Absolute value Let’s solve this Let’s solve this

Solving Inequalities This process however is only valid with the absolute value is by itself This process however is only valid with the absolute value is by itself

Process Remove all the outside components until the absolute value is alone Split the equation into two: one with a positive and the same sign the other negative with the different sign Solve as usual Figure out if this is an AND or OR statement Verify that the answer makes sense

Solving Inequalities Another example Another example

Examples

Examples

Examples

Examples

Examples

Practical Example The acceptable amount of mercury in water is 500 ppm with an absolute deviation of at most 30 ppm. What are the acceptable amounts of mercury found in water?

Most Important Points What’s the most important thing that we can learn from today? We can solve absolute value equalities by solving them twice, once for the positive value with the same inequality sign, once for the negative value and the opposite inequality sign

Homework 6.6 1, 2, 3-21 odd, 22-28, 35-38, even