^ Solve ∣3x+5∣ = ∣4x−3∣ where x ∈ R


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Question Number 109004 by Don08q last updated on 20/Aug/20

$\overset{}{} S o l v e ∣ 3 x + 5 ∣ = ∣ 4 x - 3 ∣$ $w h e r e x \in R$
	Answered by bemath last updated on 20/Aug/20

	$\Rightarrow (3 x + 5 + 4 x - 3) (3 x + 5 - 4 x + 3) = 0$ $(7 x + 2) (8 - x) = 0 \to {\begin{cases} x = 8 \\ x = - \frac{2}{7} \end{cases}$ $c h e k i n g \to {\begin{cases} ∣ 24 + 5 ∣=∣ 32 - 3 ∣ ⟨ t r u e ⟩ \\ ∣ \frac{- 6 + 35}{7} ∣ = ∣ \frac{- 8 - 21}{7} ∣ ⟨ t r u e ⟩ \end{cases}$
	Commented by john santu last updated on 20/Aug/20

	$s h o r t c u t$
	Answered by Aziztisffola last updated on 20/Aug/20

	$∣ 3 x + 5 ∣ = ∣ 4 x - 3 ∣\Rightarrow {\begin{cases} 3 x + 5 = 4 x - 3 \\ 3 x + 5 = - 4 x + 3 \end{cases}$ ${\begin{cases} x = 8 \\ x = - \frac{2}{7} \end{cases}$
	Commented by Rasheed.Sindhi last updated on 21/Aug/20

	$S t a n d a r d w a y!$
	Answered by floor(10²Eta[1]) last updated on 20/Aug/20

	$squaring both sides :$ ${9 x}^{2} + 30 x + 25 = {16 x}^{2} - 24 x + 9$ ${7 x}^{2} - 54 x - 16 = 0$ $x = \frac{54 \pm \sqrt{54^{2} + 4.7.16}}{2.7} = \frac{27 \pm 29}{7}$ $\Rightarrow x = \frac{- 2}{7}, x = 8$
	Answered by Rasheed.Sindhi last updated on 21/Aug/20

	$∣ 3 x + 5 ∣ = ∣ 4 x - 3 ∣\Rightarrow \frac{∣ 3 x + 5 ∣}{∣ 4 x - 3 ∣} = 1$ $\Rightarrow∣ \frac{3 x + 5}{4 x - 3} ∣= 1$ $\Rightarrow \frac{3 x + 5}{4 x - 3} = 1 \lor \frac{3 x + 5}{4 x - 3} = - 1$ $4 x - 3 = 3 x + 5 \lor 3 x + 5 = - 4 x + 3$ $x = 8 \lor x = - 2 / 7$
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