Question

Which of the following number lines and solution sets show the values of r that make the inequality -2r+3<9 true

Answer 1

Answer:

"Solving'' an inequality means finding all of its solutions. A "solution'' of an inequality is a number which when substituted for the variable makes the inequality a true statement. When we substitute 8 for x, the inequality becomes 8-2 > 5. Thus, x=8 is a solution of the inequality.

Step-by-step explanation:

Example 1:

Consider the inequality

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The basic strategy for inequalities and equations is the same: isolate x on one side, and put the "other stuff" on the other side. Following this strategy, let's move +5 to the right side. We accomplish this by subtracting 5 on both sides (Rule 1) to obtain

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after simplification we obtain

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Once we divide by +2 on both sides (Rule 3a), we have succeeded in isolating x on the left:

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or simplified,

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All real numbers less than 1 solve the inequality. We say that the "set of solutions'' of the inequality consists of all real numbers less than 1. In interval notation, the set of solutions is the interval tex2html_wrap_inline187 .

Example 2:

Find all solutions of the inequality

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Let's start by moving the ``5'' to the right side by subtracting 5 on both sides (Rule 1):

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or simplified,

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How do we get rid of the ``-'' sign in front of x? Just multiply by (-1) on both sides (Rule 3b), changing " tex2html_wrap_inline201 " to " tex2html_wrap_inline203 " along the way:

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or simplified

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All real numbers greater than or equal to -1 satisfy the inequality. The set of solutions of the inequality is the interval tex2html_wrap_inline205 .

Example 3:

Solve the inequality

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Let us simplify first:

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There is more than one route to proceed; let's take this one: subtract 2x on both sides (Rule 1).

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and simplify:

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Next, subtract 9 on both sides (Rule 1):

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simplify to obtain

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Then, divide by 4 (Rule 3a):

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and simplify again:

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It looks nicer, if we switch sides (Rule 2).

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In interval notation, the set of solutions looks like this: tex2html_wrap_inline227 .