Solve for j |9-2j|+10>=2

$| 9 - 2 j | + 10 \geq 2$

Step 1

Write $| 9 - 2 j | + 10 \geq 2$ as a piecewise.

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$9 - 2 j \geq 0$

Solve the inequality.

Subtract $9$ from both sides of the inequality.

$- 2 j \geq - 9$

Divide each term in $- 2 j \geq - 9$ by $- 2$ and simplify.

Divide each term in $- 2 j \geq - 9$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 j}{- 2} \leq \frac{- 9}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 j}{- 2} \leq \frac{- 9}{- 2}$

Divide $j$ by $1$ .

$j \leq \frac{- 9}{- 2}$

Simplify the right side.

Dividing two negative values results in a positive value.

$j \leq \frac{9}{2}$

In the piece where $9 - 2 j$ is non-negative, remove the absolute value.

$9 - 2 j + 10 \geq 2$

To find the interval for the second piece, find where the inside of the absolute value is negative.

$9 - 2 j < 0$

Solve the inequality.

Subtract $9$ from both sides of the inequality.

$- 2 j < - 9$

Divide each term in $- 2 j < - 9$ by $- 2$ and simplify.

Divide each term in $- 2 j < - 9$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 j}{- 2} > \frac{- 9}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 j}{- 2} > \frac{- 9}{- 2}$

Divide $j$ by $1$ .

$j > \frac{- 9}{- 2}$

Simplify the right side.

Dividing two negative values results in a positive value.

$j > \frac{9}{2}$

In the piece where $9 - 2 j$ is negative, remove the absolute value and multiply by $- 1$ .

$- (9 - 2 j) + 10 \geq 2$

Write as a piecewise.

${\begin{cases} 9 - 2 j + 10 \geq 2 & j \leq \frac{9}{2} \\ - (9 - 2 j) + 10 \geq 2 & j > \frac{9}{2} \end{cases}$

Add $9$ and $10$ .

${\begin{cases} - 2 j + 19 \geq 2 & j \leq \frac{9}{2} \\ - (9 - 2 j) + 10 \geq 2 & j > \frac{9}{2} \end{cases}$

Simplify $- (9 - 2 j) + 10 \geq 2$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} - 2 j + 19 \geq 2 & j \leq \frac{9}{2} \\ - 1 \cdot 9 - (- 2 j) + 10 \geq 2 & j > \frac{9}{2} \end{cases}$

Multiply $- 1$ by $9$ .

${\begin{cases} - 2 j + 19 \geq 2 & j \leq \frac{9}{2} \\ - 9 - (- 2 j) + 10 \geq 2 & j > \frac{9}{2} \end{cases}$

Multiply $- 2$ by $- 1$ .

${\begin{cases} - 2 j + 19 \geq 2 & j \leq \frac{9}{2} \\ - 9 + 2 j + 10 \geq 2 & j > \frac{9}{2} \end{cases}$

Add $- 9$ and $10$ .

${\begin{cases} - 2 j + 19 \geq 2 & j \leq \frac{9}{2} \\ 2 j + 1 \geq 2 & j > \frac{9}{2} \end{cases}$

Step 2

Solve $- 2 j + 19 \geq 2$ when $j \leq \frac{9}{2}$ .

Solve $- 2 j + 19 \geq 2$ for $j$ .

Move all terms not containing $j$ to the right side of the inequality.

Subtract $19$ from both sides of the inequality.

$- 2 j \geq 2 - 19$

Subtract $19$ from $2$ .

$- 2 j \geq - 17$

Divide each term in $- 2 j \geq - 17$ by $- 2$ and simplify.

Divide each term in $- 2 j \geq - 17$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 j}{- 2} \leq \frac{- 17}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 j}{- 2} \leq \frac{- 17}{- 2}$

Divide $j$ by $1$ .

$j \leq \frac{- 17}{- 2}$

Simplify the right side.

Dividing two negative values results in a positive value.

$j \leq \frac{17}{2}$

Find the intersection of $j \leq \frac{17}{2}$ and $j \leq \frac{9}{2}$ .

$j \leq \frac{9}{2}$

Step 3

Solve $2 j + 1 \geq 2$ when $j > \frac{9}{2}$ .

Solve $2 j + 1 \geq 2$ for $j$ .

Move all terms not containing $j$ to the right side of the inequality.

Subtract $1$ from both sides of the inequality.

$2 j \geq 2 - 1$

Subtract $1$ from $2$ .

$2 j \geq 1$

Divide each term in $2 j \geq 1$ by $2$ and simplify.

Divide each term in $2 j \geq 1$ by $2$ .

$\frac{2 j}{2} \geq \frac{1}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 j}{2} \geq \frac{1}{2}$

Divide $j$ by $1$ .

$j \geq \frac{1}{2}$

Find the intersection of $j \geq \frac{1}{2}$ and $j > \frac{9}{2}$ .

$j > \frac{9}{2}$

Step 4

Find the union of the solutions for any value of $j$ .

All real numbers

Step 5

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$

Step 6

Solve for j |9-2j|+10>=2

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