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How many different values of x are there such that |x + 10| = 2x + 8?
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13 May 2024, 01:30
13 May 2024, 01:30
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How many different values of \(x\) are there such that \(|x+10|=2x+8\)?
A. None
B. 1
C. 2
D. 3
E. Infinitely many
A. None
B. 1
C. 2
D. 3
E. Infinitely many
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Re: How many different values of x are there such that |x + 10| = 2x + 8?
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01 Jun 2024, 07:15
01 Jun 2024, 07:15
Expert Reply
Official Solution:
How many different values of \(x\) are there such that \(|x+10|=2x+8\)?
A. None
B. 1
C. 2
D. 3
E. Infinitely many
The left-hand side of the equation is an absolute value, which is always non-negative. Thus, the right-hand side must also be non-negative: \(2x+8 \ge 0\), which gives us \(x \ge -4\). For this range, \(x+10\) is positive, so \(|x+10|=x+10\). Therefore, \(|x+10|=2x+8\) can be rewritten as \(x+10=2x+8\). Solving for \(x\) gives \(x=2\). Therefore, only one value of \(x\) satisfies \(|x+10|=2x+8\).
Answer: B
How many different values of \(x\) are there such that \(|x+10|=2x+8\)?
A. None
B. 1
C. 2
D. 3
E. Infinitely many
The left-hand side of the equation is an absolute value, which is always non-negative. Thus, the right-hand side must also be non-negative: \(2x+8 \ge 0\), which gives us \(x \ge -4\). For this range, \(x+10\) is positive, so \(|x+10|=x+10\). Therefore, \(|x+10|=2x+8\) can be rewritten as \(x+10=2x+8\). Solving for \(x\) gives \(x=2\). Therefore, only one value of \(x\) satisfies \(|x+10|=2x+8\).
Answer: B
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Re: How many different values of x are there such that |x + 10| = 2x + 8?
[#permalink]
13 May 2024, 02:10
13 May 2024, 02:10
1
Kudos
Bunuel wrote:
How many different values of \(x\) are there such that \(|x+10|=2x+8\)?
A. None
B. 1
C. 2
D. 3
E. Infinitely many
A. None
B. 1
C. 2
D. 3
E. Infinitely many
\(|x+10|=2x+8\)
As the RHS of the equation is modulus, \(2x + 8 \geq 0\)
⇒ \(x \geq -4\)
\(|x+10|=2x+8\)
Squaring both sides we get
\(x^2 + 100 + 20x = 4x^2 + 64 + 32x\)
\(3x^2 + 12x - 36 = 0\)
\(x^2 + 4x - 12 = 0\)
\(x + 6x - 2x - 12 = 0\)
\((x + 6)(x - 2) = 0\)
Roots = \(-6, +2\)
However, the value of \(x\) must be greater than \(-4.\) Hence, only one value, \(+2\), holds in this case.
Option B
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