QuestionFind the graph of the inequality $|\frac{1}{2} x + 2| \geq 1$ .

Studdy Solution

STEP 1

Assumptions1. The equation is $\left|\frac{1}{} x+\right| \geq1$ . We need to find the graph of this equation

STEP 2

We first need to understand what the absolute value function is. The absolute value of a number is its distance from zero on the number line, regardless of direction. In other words, it makes any negative number positive.

STEP 3

The absolute value inequality $\left|\frac{1}{2} x+2\right| \geq1$ can be split into two separate inequalities1. $\frac{1}{2} x+2 \geq1$ (for the positive solution)
2. $-\left(\frac{1}{2} x+2\right) \geq1$ (for the negative solution)

STEP 4

We can solve the first inequality, $\frac{1}{2} x+2 \geq1$ , by subtracting2 from both sides.
$\frac{1}{2} x \geq1 -2$

STEP 5

implify the right side of the inequality.
$\frac{1}{2} x \geq -1$

STEP 6

To isolate $x$ , multiply both sides of the inequality by2.
$x \geq -1 \times2$

STEP 7

implify the right side of the inequality.
$x \geq -2$

STEP 8

Now, let's solve the second inequality, $-\left(\frac{1}{2} x+2\right) \geq1$ , by distributing the negative sign.
$-\frac{1}{2} x -2 \geq1$

STEP 9

Add2 to both sides of the inequality to isolate the term with $x$ .
$-\frac{}{2} x \geq +2$

STEP 10

implify the right side of the inequality.
$-\frac{}{2} x \geq3$

STEP 11

To isolate $x$ , divide both sides of the inequality by $-\frac{}{}$ . Remember that when we divide or multiply an inequality by a negative number, the direction of the inequality sign changes.
$x \leq \frac{3}{-\frac{}{}}$

STEP 12

implify the right side of the inequality.
$x \leq -6$

STEP 13

Now we have two inequalities $x \geq -2$ and $x \leq -6$ .

STEP 14

Plot these inequalities on a number line. The solution to the original absolute value inequality is the intersection of these two inequalities. In this case, there is no intersection, so the solution set is empty.

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