Answer: " w = (all real numbers) " —as explained below.
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Explanation:
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Given the following "inequality":
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" -3(w – 3) ≥ 9 – 3w " ; Solve for "w" ;
and see if the answer ["value for "w"]; is:
"all real numbers."
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→ " -3(w – 3) ≥ 9 – 3w " ;
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Method 1):
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On the "right-hand side" of the "inequality";
Factor out a "(-3)" ;
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" -3(w – 3) ≥ ( -3 * ? = 9?) – (-3 * ? = 3w? ) " ;
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1) " -3 * (what value?) = 9 "?
Let "x" be the 'unknown value' :
→ " -3x = 9 " ;
Divide each side of the equation by "(-3)" ;
to isolate "x" on one of the equation;
& to solve for "x" ;
-3x / -3 = 9 / -3 ;
to get: " x = -3 " ;
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2) " -3 * (what value?) = 3w " ?
Let "x" be the 'unknown value' :
→ " -3x = 3w " ? ;
→ Divide each side or the equation by ("-3").
→ to isolate "x" on one side of the equation;
→ & to solve for "x" ;
-3x / 3 = 3w / -3 ;
to get: x = -1w ;
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So: " -3(w – 3) ≥ -3* (? = 9?) – (-3 * ? = 3w?) " ;
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Rewrite as:
" -3(w – 3) ≥ -3* [(-3 – (-1w)] " ;
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Now, let us consider the Following Portion of the "right-hand side" of the inequality:;
" (-3 – (-1w)] = " (-3 + 1w) " ;
→ {since: "subtracting a "negative value is the equivalent of adding that particular value's positive value"} ;
and bring down the "(-3)" on the "right-hand side" of the inequality; and rewrite the inequality as:
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" -3(w – 3) ≥ -3(-3 + 1w) " ;
→ Now, divide EACH SIDE of the inequality by "(-3)" :
{Note: Each time when one multiplies or divides an inequality by a "negative value"—the inequality sign flips to the other direction.}.
→ [ -3(w – 3)] / -3 ≥ [-3(-3 + 1w ] / -3 ;
to get:
→ " (w – 3) ≤ (-3 + 1w) " ;
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Note: " (-3 + 1w) " ; ↔ " [(1w + (-3)] = " 1w – 3 " = "w – 3 " ;
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Rewrite the inequality:
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→ " (w – 3) ≤ (w – 3 ) " ;
↔ " w – 3 ≤ w – 3 " ;
The same value is equal to each other: not less than or equal to each other: For instance:
→ " w – 3 ≤ w – 3 " ;
If we add "3" to each side of the equation:
→ " w – 3 + 3 ≤ w – 3 + 3 " ;
We get: " w ≤ w " . "w = w". "w is not "less than" itself. So all real numbers apply!
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Method 2)
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Given the following "inequality":
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" -3(w – 3) ≥ 9 – 3w " ; Solve for "w" ;
and see if the answer ["value for "w"]; is:
"all real numbers."
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On the "left-hand side of the inequality;
we have:
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" -3(w – 3) " ;
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Take note of the "distributive property of multiplication" :
→ " a(b + c) = ab + ac " ;
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As such:
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We can expand:
" -3(w – 3) = (-3*w) + (-3*-3) " ;
= -3w + (9) ;
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Now, we can rewrite the original inequality:
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" -3w + 9 ≥ 9 – 3w " ;
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Note: " -3w + 9 = 9 + (-3w) = 9 – 3w " ;
→ {since: adding a "negative value" gets the same value as subtracting that value's "positive equivalent"};
And we can rewrite our inequality as:
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" 9 – 3w ≥ 9 – 3w " ;
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Note: We have the same value on each side.
"(9 – 3w)" is not greater than itself; it is "equal to itself".
Any and all real numbers as values for "w" will result in the same value for any particular expression's Exact Same Expression!
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So: "w = (all real numbers)" .
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Hope this is helpful to you!
Best wishes!
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