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3 years ago

Solve 8 − |2x − 1| ≥ 6. State your answer in interval notation

Mathematics

1 answer:

Gnoma [55]3 years ago

3 0

Answer:

$\huge\boxed{x\in[-0.5,\ 1.5]}$

Step-by-step explanation:

$8-|2x-1|\geq6\qquad\text{subtract 8 from both sides}\\\\8-8-|2x-1|\geq6-8\\\\-|2x-1|\geq-2\qquad\text{change the signs}\ /\text{multiply both sides by (-1)}/\\\\|2x-1|\leq2\iff2x-1\leq2\ \wedge\ 2x-1\geq-2\qquad\text{add 1 to all sides}\\\\2x-1+1\leq2+1\ \wedge\ 2x-1+1\geq-2+1\\\\2x\leq3\ \wedge\ 2x\geq-1\qquad\text{divide all sides by 2}\\\\\dfrac{2x}{2}\leq\dfrac{3}{2}\ \wedge\ \dfrac{2x}{2}\geq\dfrac{-1}{2}\\\\x\leq1.5\ \wedge\ x\geq-0.5\Rightarrow-0.5\leq x\leq1.5$

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Hey guys! Could someone help me with this problem? I got the wrong answer! The answer is B, but I don’t know how to get it! Coul

asambeis [7]

When it says 'which has a greater rate of change' it is referring to 'what has a greater slope.' So, let's calculate each thing's slope:

Table:
I'm going to pick the easiest values to work with, so I'll pick the points (0, -2) and (1, -9/5) - I simplified the mixed fraction. So, here we go, the slope formula is:
$m =\frac{y_2-y_1}{x_2-x_1}$
$m = \frac{- \frac{9}{5} - (-2)}{1-0} = \frac{- \frac{9}{5} - ( -\frac{10}{5} )}{1-0}$
$m = \frac{ \frac{1}{5} }{1} = \frac{1}{5}$

So, the table's slope is -1/5.

Graph
We can, again, pick two points. I'll pick the two given: (5, 0) and (0, -3). Let's start off with the same formula:
$m =\frac{y_2-y_1}{x_2-x_1}$
$m = \frac{-3-0}{0-5} = \frac{3}{5}$

The graph's slope is 3/5.

Now, we just have to decide which is bigger: -1/5 or 3/5. So, clearly:
3/5 > -1/5.

The 3/5 slope was the graph, hence it has the greater rate of change (that's why it's B). If anything doesn't make sense, feel free to DM me or ask!

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