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Solve each inequality or compound inequality. Write the solution set in interval notation and graph it.$$-18 \leq 9(x-5)<27$$
[3,8)
Precalculus
Algebra
Chapter 2
Equations, Inequalities, and Problem Solving
Section 7
Solving Inequalities
Algebra Topics That are Reviewed at the Start of the Semester
Equations and Inequalities
Campbell University
University of Michigan - Ann Arbor
Idaho State University
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solve the compound inequality, right? The solution set an interval notation, then Graff it. We have the compound. Inequality of negative 18 is less than or equal to the product of nine, and the binomial X minus five is less than 27. So to solve the compound inequality, we want to use properties of inequality toe. Isolate the variable as the middle term of the inequality. The first thing we want to d'oh is rewrite the inequality without parentheses. I'm noticing that each part of the inequality negative 18 9 times X minus five and 27 are all divisible by nine. So instead of distributing the nine into the binomial, let's go ahead and divide all parts of that inequality by nine instead. So now we have negative 18. Divided by nine is negative, too, is less than or equal to nine times X, minus five, divided by nine. And what happens here is the nine cancels the other nine as a common factor, leaving us with just X minus five as our middle term of the inequality and then is less than 27 divided by nine, which is three. Our middle term is now X minus five toe isolate X. We need to undo the subtraction using the inverse operation addition. So let's add five to all parts of the inequality, which will leave us with positive. Three is less than or equal to X is less than eight. This compound inequality can be represented on a number line to show that the solution set is all values that represent X being greater than or equal to three, but less than eight. So we'll start at three with a closed circle because three is included in the solution set and we will end at positive eight with an open circle to represent The X is not part of this, uh, equal to eight, so it is not part of the solution set. Then we're going to highlight the solution set between the two end points of three and eight. To represent the entire solution. Set an interval notation. We will open with a bracket at positive three and close with a parentheses at eight
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