Question 333·Hard·Math

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For which value of $m$ does the equation

(m+2)(4x-7)=4(m+2)x+9

have infinitely many real solutions for $x$ ?

The phrase infinitely many solutions signals an identity: a linear equation that is true for every $x$ . Distribute, then compare the $x$ -terms and the terms without $x$ . If the $x$ -terms already match, a Desmos regression can find the value that makes the remaining terms match too. Setting a shared factor to zero does not erase a number added outside it.

Hints

Hint 1
For an identity, every value of $x$ must work. Distribute $(m+2)$ across both terms in $4x-7$ . How does the resulting $x$ -term compare with the one on the right?
Hint 2
The constant terms are the parts without $x$ . Once the matching $x$ -terms cancel, those constants must be equal. Keep the $+9$ on the right when you write that condition.

Step-by-step

Cancel the matching terms, then solve
Step 1
Distribute on the left
Distribute means multiply each term inside the parentheses by $(m+2)$ :
$(m+2)(4x-7)=4(m+2)x-7(m+2).$
So the given equation becomes
$4(m+2)x-7(m+2)=4(m+2)x+9.$
The $+9$ is outside the factor $(m+2)$ , so making that factor zero would not erase it.
Step 2
Match what remains after the $x$ -terms cancel
Infinitely many solutions means every real $x$ must make the equation true. The $x$ -terms are already identical. Subtract $4(m+2)x$ from both sides:
$-7(m+2)=9.$
When the $x$ -terms cancel, the remaining constants must agree. If they do not, no value of $x$ works.
Step 3
Find the value of $m$
Type $-7(m_{1}+2)\sim9$ in Desmos. The $\sim$ makes this a regression, asking Desmos to find $m_1$ rather than graph an equation. Under PARAMETERS, it shows $m_1\approx-3.285714$ . Type $m_1$ on the next line and use its fraction button to see the exact value $-\frac{23}{7}$ . At that value, both sides agree for every real $x$ . Choice D.

Question 333·Hard·Math

Strategy

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Step-by-step

Lessons that teach this

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Strategy

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Step-by-step

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