Question 1209903: Find the number of points of intersection between the graphs of the following equations: \begin{align*}
y = |2x + 5|,
y = -|3x - 2| + |4x - 7| + x.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's analyze the given equations to find the number of intersection points.
**Equation 1:** $y = |2x + 5|$
**Equation 2:** $y = -|3x - 2| + |4x - 7| + x$
We need to consider cases based on the values of $x$ that make the expressions inside the absolute values zero.
**Critical Points:**
* $2x + 5 = 0 \Rightarrow x = -5/2 = -2.5$
* $3x - 2 = 0 \Rightarrow x = 2/3$
* $4x - 7 = 0 \Rightarrow x = 7/4 = 1.75$
**Case 1: $x \le -5/2$**
* $|2x + 5| = -(2x + 5) = -2x - 5$
* $|3x - 2| = -(3x - 2) = -3x + 2$
* $|4x - 7| = -(4x - 7) = -4x + 7$
Equation 1: $y = -2x - 5$
Equation 2: $y = -(-3x + 2) + (-4x + 7) + x = 3x - 2 - 4x + 7 + x = 5$
Set the equations equal:
$-2x - 5 = 5$
$-2x = 10$
$x = -5$
Since $-5 \le -2.5$, this is a valid solution.
Intersection point: $(-5, 5)$
**Case 2: $-5/2 < x \le 2/3$**
* $|2x + 5| = 2x + 5$
* $|3x - 2| = -(3x - 2) = -3x + 2$
* $|4x - 7| = -(4x - 7) = -4x + 7$
Equation 1: $y = 2x + 5$
Equation 2: $y = -(-3x + 2) + (-4x + 7) + x = 3x - 2 - 4x + 7 + x = 5$
Set the equations equal:
$2x + 5 = 5$
$2x = 0$
$x = 0$
Since $-2.5 < 0 \le 2/3$, this is a valid solution.
Intersection point: $(0, 5)$
**Case 3: $2/3 < x \le 7/4$**
* $|2x + 5| = 2x + 5$
* $|3x - 2| = 3x - 2$
* $|4x - 7| = -(4x - 7) = -4x + 7$
Equation 1: $y = 2x + 5$
Equation 2: $y = -(3x - 2) + (-4x + 7) + x = -3x + 2 - 4x + 7 + x = -6x + 9$
Set the equations equal:
$2x + 5 = -6x + 9$
$8x = 4$
$x = 1/2 = 0.5$
However, $2/3 \approx 0.666$, so $0.5$ is not in the interval, so no solution in this case.
**Case 4: $x > 7/4$**
* $|2x + 5| = 2x + 5$
* $|3x - 2| = 3x - 2$
* $|4x - 7| = 4x - 7$
Equation 1: $y = 2x + 5$
Equation 2: $y = -(3x - 2) + (4x - 7) + x = -3x + 2 + 4x - 7 + x = 2x - 5$
Set the equations equal:
$2x + 5 = 2x - 5$
$5 = -5$
This is a contradiction, so there are no solutions in this case.
**Intersection Points:**
* $(-5, 5)$
* $(0, 5)$
There are two intersection points.
Final Answer: The final answer is $\boxed{2}$
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