Question

9)sketch the graph of y = -3|x - 2|

1. which pair of linear equations would be used to solve the absolute value equation |x + 1| - 2 = 6?

a) x + 1 = 4 or x + 1 = -4
b) x + 1 = 8 or x + 1 = -8
c) x - 1 = 8 or x - 1 = -8
d) x + 1 = 6 or x + 1 = -6

1. solve the absolute value equation: |x| + 3 = 10

Explanation:

Question 9

Step1: Recall absolute value graph form

The parent function is \( y = |x| \), which is a V - shaped graph with vertex at \((0,0)\), opening upwards. The function \( y=-3|x - 2|\) is a transformation of \( y = |x|\). The general form of an absolute - value function is \( y=a|x - h|+k \), where \((h,k)\) is the vertex, \(a\) determines the vertical stretch/compression and reflection.
For \( y=-3|x - 2|\), \(h = 2\), \(k = 0\), so the vertex is at \((2,0)\). The coefficient \(a=-3\), the negative sign means the graph is reflected over the \(x\) - axis (opens downwards), and the factor of 3 means it is vertically stretched by a factor of 3.

Step2: Find some points

• When \(x = 2\), \(y=-3|2 - 2|=0\) (vertex point).
• When \(x=1\), \(y=-3|1 - 2|=-3|-1|=-3\). So the point is \((1, - 3)\).
• When \(x = 3\), \(y=-3|3 - 2|=-3|1|=-3\). So the point is \((3,-3)\).
• When \(x = 0\), \(y=-3|0 - 2|=-3| - 2|=-6\). So the point is \((0,-6)\).
• When \(x = 4\), \(y=-3|4 - 2|=-3|2|=-6\). So the point is \((4,-6)\).

Plot the vertex \((2,0)\) and the other points \((1, - 3)\), \((3,-3)\), \((0,-6)\), \((4,-6)\) and draw a V - shaped graph (opening downwards) passing through these points.

Question 10

Step1: Isolate the absolute value

Start with the equation \(|x + 1|-2=6\). Add 2 to both sides of the equation to isolate the absolute - value term.
Using the addition property of equality: \(|x + 1|-2 + 2=6 + 2\), which simplifies to \(|x + 1|=8\).

Step2: Apply the absolute - value property

The absolute - value property states that if \(|A|=b\) (where \(b\geq0\)), then \(A = b\) or \(A=-b\).
In our case, \(A=x + 1\) and \(b = 8\). So we have \(x + 1=8\) or \(x + 1=-8\).

Step1: Isolate the absolute value

Start with the equation \(|x|+3 = 10\). Subtract 3 from both sides of the equation.
Using the subtraction property of equality: \(|x|+3-3=10 - 3\), which gives \(|x|=7\).

Step2: Solve for \(x\) using absolute - value property

By the absolute - value property, if \(|x|=7\), then \(x = 7\) or \(x=-7\) (because the absolute value of a number is its distance from 0 on the number line, so both 7 and - 7 are 7 units away from 0).

Answer:

B. \(x + 1=8\) or \(x + 1=-8\)

Question 11