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masha68 [24]
2 years ago
5

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

Mathematics
1 answer:
scoray [572]2 years ago
4 0

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

displaymath173

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

displaymath174

after simplification we obtain

displaymath175

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

displaymath176

or simplified,

displaymath177

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

displaymath189

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

displaymath190

or simplified,

displaymath191

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:

displaymath192

or simplified

displaymath193

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

displaymath207

Let us simplify first:

displaymath208

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

displaymath209

and simplify:

displaymath210

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

displaymath211

simplify to obtain

displaymath212

Then, divide by 4 (Rule 3a):

displaymath213

and simplify again:

displaymath214

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

displaymath215

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

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