1) 5|x| + 3 = 7; 2) 1, 7|x| + 4, 9 = 10; 3) 2, 5|3y| = 15; 4) |2x + 14| = 6; 5) |9x + 15| = 6; 6) |8x - 6| = 14; 7) |x| - 3 = 0; 8) 5 - |x| = 0; 9) |x| + 3 = 0; 10) |7 - |x|| = 0.
Understand the Problem
The question consists of multiple mathematical equations, primarily involving absolute values and fractions, which need to be solved for the variables x and y.
Answer
1) $x = \frac{4}{5}, -\frac{4}{5}$; 2) $x = 3, -3$; 3) $y = 2, -2$; 4) $x = -4, -10$; 5) $x = -1, -\frac{7}{3}$; 6) $x = \frac{5}{2}, -1$; 7) $x = 3, -3$; 8) $x = 5, -5$; 9) No solution; 10) $x = 7, -7$.
Answer for screen readers
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$x = \frac{4}{5}, -\frac{4}{5}$
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$x = 3, -3$
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$y = 2, -2$
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$x = -4, -10$
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$x = -1, -\frac{7}{3}$
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$x = \frac{5}{2}, -1$
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$x = 3, -3$
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$x = 5, -5$
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No solution
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$x = 7, -7$
Steps to Solve
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Solve each equation separately
We will tackle each equation step-by-step.
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Equation 1: Solve $5|x| + 3 = 7$
Subtract 3 from both sides:
$$ 5|x| = 4 $$
Divide by 5:
$$ |x| = \frac{4}{5} $$
Thus, $x = \frac{4}{5}$ or $x = -\frac{4}{5}$.
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Equation 2: Solve $1.7|x| + 4.9 = 10$
Subtract 4.9 from both sides:
$$ 1.7|x| = 5.1 $$
Divide by 1.7:
$$ |x| = 3 $$
Thus, $x = 3$ or $x = -3$.
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Equation 3: Solve $2.5|3y| = 15$
Divide both sides by 2.5:
$$ |3y| = 6 $$
Divide by 3:
$$ |y| = 2 $$
Thus, $y = 2$ or $y = -2$.
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Equation 4: Solve $|2x + 14| = 6$
This splits into two cases:
Case 1: $2x + 14 = 6$
Subtract 14:
$$ 2x = -8 $$
Divide by 2:
$$ x = -4 $$
Case 2: $2x + 14 = -6$
Subtract 14:
$$ 2x = -20 $$
Divide by 2:
$$ x = -10 $$
So, $x = -4$ or $x = -10$.
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Equation 5: Solve $|9x + 15| = 6$
This splits into two cases:
Case 1: $9x + 15 = 6$
Subtract 15:
$$ 9x = -9 $$
Divide by 9:
$$ x = -1 $$
Case 2: $9x + 15 = -6$
Subtract 15:
$$ 9x = -21 $$
Divide by 9:
$$ x = -\frac{7}{3} $$
So, $x = -1$ or $x = -\frac{7}{3}$.
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Equation 6: Solve $|8x - 6| = 14$
This splits into two cases:
Case 1: $8x - 6 = 14$
Add 6:
$$ 8x = 20 $$
Divide by 8:
$$ x = \frac{5}{2} $$
Case 2: $8x - 6 = -14$
Add 6:
$$ 8x = -8 $$
Divide by 8:
$$ x = -1 $$
So, $x = \frac{5}{2}$ or $x = -1$.
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Equation 7: Solve $|x| - 3 = 0$
Add 3 to both sides:
$$ |x| = 3 $$
Thus, $x = 3$ or $x = -3$.
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Equation 8: Solve $5 - |x| = 0$
Add $|x|$ to both sides:
$$ 5 = |x| $$
Thus, $x = 5$ or $x = -5$.
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Equation 9: Solve $|x| + 3 = 0$
Subtract 3 from both sides:
$$ |x| = -3 $$
This has no solution since absolute values cannot be negative.
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Equation 10: Solve $|7 - |x|| = 0$
This implies:
$$ 7 - |x| = 0 $$
Thus,
$$ |x| = 7 $$
So, $x = 7$ or $x = -7$.
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$x = \frac{4}{5}, -\frac{4}{5}$
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$x = 3, -3$
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$y = 2, -2$
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$x = -4, -10$
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$x = -1, -\frac{7}{3}$
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$x = \frac{5}{2}, -1$
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$x = 3, -3$
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$x = 5, -5$
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No solution
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$x = 7, -7$
More Information
Absolute values always yield non-negative results, which is crucial in determining the potential solutions for equations involving them.
Tips
- Forgetting to consider both cases when solving absolute value equations.
- Miscalculating negative cases or failing to combine results from both scenarios.
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