If \(\frac{3c^{2}}{6c}\)+ 9 = 15, what is the value of c?
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Solution
First, reduce the fraction \(\frac{(3c^{2})}{(6c)}\) by dividing the numerator and denominator by 3c;\(\frac{(3c^{2})}{(6c)}\) =c⁄2.Now, subtract 9 from both sides of the equation and then multiply both sides of the equation by 2:
c⁄2 + 9 = 15
c⁄2 + 9 – 9 = 15 – 9
c⁄2 = 6
(2)(c⁄2) = (6)(2)
c = 12
What is the solution set for the inequality –8(x + 3) ≤ 2(–2x + 10)?
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Solution
First, multiply (x + 3) by –8 and multiply (–2x + 10) by 2: –8(x + 3) = –8x – 24, 2(–2x + 10) = –4x + 20. Then, add 8x to both sides of the inequality and subtract 20 from both sides of the inequality:
–8x – 24 ≤ –4x + 20
–8x – 24 + 8x ≤ –4x + 20 + 8x
–24 ≤ 4x + 20
–24 – 20 ≤ 4x + 20 – 20
–44 ≤ 4x
Finally, divide both sides of the inequality by 4:–\(\frac{44}{4}\leq \frac{4x}{4}\), x ≥ –11.
If 6x – 4x + 9 = 6x + 4 – 9, what is the value of x?
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Solution
First, combine like terms on each side of the equation; 6x – 4x = 2x and 4 – 9 = –5. Now, subtract 2x from both sides of the equation and add 5 to both sides of the equation:
2x + 9 = 6x – 5
2x – 2x + 9 = 6x – 2x – 5
9 = 4x – 5
9 + 5 = 4x – 5 + 5
14 = 4x
Finally, divide both sides of the equation by 4:\(\frac{14}{4}=\frac{4x}{4}\), x = \(\frac{14}{4}\) = 7⁄2.
The inequality 3x – 6 ≤ 4(x + 2) is equivalent to
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Solution
First, multiply (x + 2) by 4: 4(x + 2) = 4x + 8. Then, subtract 3x from both sides of the inequality and subtract 8 from both sides of the inequality:
3x – 6 ≤ 4x + 8
3x – 6 – 3x ≤ 4x + 8 – 3x
–6 ≤ x + 8
–6 – 8 ≤ x + 8 – 8
x ≥ –14
If 9a + 5 = –22, what is the value of a?
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Solution
To solve the equation, subtract 5 from both sides of the equation, then divide by 9: 9a + 5 = –22, 9a + 5 – 5 = –22 – 5, 9a = –27, a = –3.
The inequality –3n < 12 is equivalent to
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Solution
To solve the inequality, divide both sides of the inequality by –3: –3n < 12, \(\frac{-3n}{-3}\) > \(\frac{-3n}{-3}\). Remember, when multiplying or dividing both sides of an inequality by a negative number, you must reverse the inequality symbol; n > –4.
If k⁄8 = 8, k =
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Solution
To solve the equation, multiply both sides of the equation by 8: k⁄8= 8, (8)k⁄8 = (8)(8), k = 64.
If x + 10 = 5, x =
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Solution
To solve the equation, subtract 10 from both sides of the equation: x + 10 = 5, x + 10 – 10 = 5 – 10, x = –5.
If 6p ≥ 10, then
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Solution
To solve the inequality, divide both sides of the inequality by 6: 6p ≥ 10,\(\frac{6p}{6}\) ≥\(\frac{10}{6}\), p ≥ 5⁄3.
If a – 12 = 12, a =
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Solution
To solve the equation, add 12 to both sides of the equation: a – 12 = 12, a – 12 + 12 = 12 + 12, a = 24.