PLEASE HELP!!! 100 POINTS AND IF YOU SHOW ALL STEPS AND EXPLAIN ILL GIVE BRAINLIST!!!! PLEASE SOON!!!
the absolute value of the quantity one fourth times x minus 2 end quantity minus 3 is greater than or equal to 4

Part A: Solve the inequality, showing all necessary steps. (3 points)
Part B: Describe the graph of the solution. (3 points)

Because an absolute value sign gets rid of any negative we may have, we must now split the inequalty into two: one where the value inside is positive, and one where the value inside is negative.

[tex]\frac{1}{4} x-2 \geq7\\\frac{1}{4} x \geq 9\\x \geq 36[/tex]

So, any x-values that are greater than or equal to 36 will make this inequality true. Now let's calculate the negative value possibility.

[tex]-(\frac{1}{4} x-2) \geq 7\\-\frac{1}{4} x +2 \geq 7\\-\frac{1}{4} x \geq 5\\x \leq-20[/tex]

Remeber to switch the sign of the inequality because we multiplied by a negative value. We find that the solution to this inequality is x≤-20 ∪ x≥36.

Part B -- Describe the graph:

To graph an inequality, graph the line of its equation, then shade in the correct side. The equations of this inequality are x = -20 and x = 36, which are represented as two vertical lines at -20 and 36.

The inequalites include an 'equal-to', so the line should be solid. The line x≤-20 should be shaded on its left side and x≥36 should be shaded on its right side (just like a number line).

semsee45 semsee45 · Answer 2 · 2024-07-29T03:59:08-04:00

Answer:

A) (-∞, -20] ∪ [36, ∞)

B) See below for the description.

Step-by-step explanation:

Part A

Given absolute value inequality:

[tex]\left|\frac{1}{4}x-2\right|-3\geq 4[/tex]

To solve an inequality containing an absolute value, begin by isolating the absolute value on one side of the equation:

[tex]\left|\frac{1}{4}x-2\right|-3+3\geq 4+3\\\\\left|\frac{1}{4}x-2\right|\geq7[/tex]

Now, create two separate cases by considering both the positive and negative contents of the absolute value:

[tex]\textsf{Case 1:}\quad\;\;\;\: \left(\frac{1}{4}x-2\right)\geq7 \\\\\textsf{Case 2:}\quad -\left(\frac{1}{4}x-2\right)\geq7[/tex]

Solve each case for x, remembering that when we multiply (or divide) both sides of an inequality by a negative number, we need to reverse the direction of the inequality sign.

[tex]\boxed{\begin{array}{c|c}\underline{\sf Case\;1}&\underline{\sf Case\;2}\\\\\begin{aligned}\left(\frac14x-2\right)&\geq 7\\\\\frac14x-2&\geq 7\\\\\frac14x-2+2&\geq 7+2\\\\\frac14x&\geq 9\\\\\frac14x\cdot 4&\geq 9\cdot 4\\\\x& \geq 36\end{aligned}&\begin{aligned}-\left(\frac14x-2\right) & \geq 7 \\\\ -\dfrac{1}{4}x+2 & \geq 7 \\\\ -\dfrac{1}{4}x+2-2 & \geq 7-2 \\\\ -\dfrac{1}{4}x & \geq 5 \\\\ -\dfrac{1}{4}x \cdot (-4) & \geq 5 \cdot (-4)\\\\ x & \leq -20\end{aligned}\end{array}}[/tex]

As the two intervals do not overlap, the solution is:

[tex]x\leq -20 \textsf{ or }x \geq 36[/tex]

In interval notation, this is:

[tex](-\infty, -20] \cup [36, \infty)[/tex]

Part B

The graph of the solution on a number line consists of two segments:

The first segment representing (-∞, -20] has a solid dot at -20 with shading extending indefinitely to the left. The solid dot is used to indicate that -20 is included in the solution set.
The second segment representing [36, ∞) has a solid dot at 36 with shading extending indefinitely to the right. The solid dot is used to indicate that 36 is included in the solution set.

These two segments indicate that the solution includes all numbers less than or equal to -20 and all numbers greater than or equal to 36.