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Question Solve the inequality 7∣w−3∣+2≥307|w-3|+2 \geq 30 and determine if all real numbers are solutions or if there is no solution.

Studdy Solution

STEP 1

Assumptions1. The absolute value function ∣w−3∣|w-3| is always non-negative, i.e., ∣w−3∣≥0|w-3| \geq0 for all real numbers ww. . We are solving the inequality 7∣w−3∣+≥307|w-3|+ \geq30 for the variable ww.

STEP 2

First, we need to isolate the absolute value term. We can do this by subtracting2 from both sides of the inequality.
7∣w−∣+2−2≥30−27|w-|+2-2 \geq30-2

STEP 3

implify the inequality.
7∣w−3∣≥287|w-3| \geq28

STEP 4

Next, we divide both sides of the inequality by7 to isolate the absolute value term.
7∣w−3∣7≥287\frac{7|w-3|}{7} \geq \frac{28}{7}

STEP 5

implify the inequality.
∣w−3∣≥4|w-3| \geq4

STEP 6

The absolute value inequality ∣w−3∣≥4|w-3| \geq4 can be rewritten as two separate inequalities w−3≥4w-3 \geq4 and w−3≤−4w-3 \leq -4.

STEP 7

First, let's solve the inequality w−3≥4w-3 \geq4. We can do this by adding3 to both sides of the inequality.
w−3+3≥4+3w-3+3 \geq4+3

STEP 8

implify the inequality.
w≥7w \geq7

STEP 9

Next, let's solve the inequality w−3≤−4w-3 \leq -4. We can do this by adding3 to both sides of the inequality.
w−3+3≤−4+3w-3+3 \leq -4+3

STEP 10

implify the inequality.
w≤−w \leq -

STEP 11

So, the solution to the inequality 7∣w−3∣+≥307|w-3|+ \geq30 is w≥7w \geq7 and w≤−w \leq -. In interval notation, this is (−∞,−]∪[7,∞)(-\infty, -] \cup [7, \infty).

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