Solve for a 4|6-2a|+8<=24

$4 | 6 - 2 a | + 8 \leq 24$

Write $4 | 6 - 2 a | + 8 \leq 24$ as a piecewise.

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

$6 - 2 a \geq 0$

Solve the inequality.

Subtract $6$ from both sides of the inequality.

$- 2 a \geq - 6$

Divide each term in $- 2 a \geq - 6$ by $- 2$ and simplify.

Divide each term in $- 2 a \geq - 6$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 a}{- 2} \leq \frac{- 6}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 a}{- 2} \leq \frac{- 6}{- 2}$

Divide $a$ by $1$ .

$a \leq \frac{- 6}{- 2}$

Simplify the right side.

Divide $- 6$ by $- 2$ .

$a \leq 3$

In the piece where $6 - 2 a$ is non-negative, remove the absolute value.

$4 (6 - 2 a) + 8 \leq 24$

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

$6 - 2 a < 0$

Solve the inequality.

Subtract $6$ from both sides of the inequality.

$- 2 a < - 6$

Divide each term in $- 2 a < - 6$ by $- 2$ and simplify.

Divide each term in $- 2 a < - 6$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 a}{- 2} > \frac{- 6}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 a}{- 2} > \frac{- 6}{- 2}$

Divide $a$ by $1$ .

$a > \frac{- 6}{- 2}$

Simplify the right side.

Divide $- 6$ by $- 2$ .

$a > 3$

In the piece where $6 - 2 a$ is negative, remove the absolute value and multiply by $- 1$ .

$4 (- (6 - 2 a)) + 8 \leq 24$

Write as a piecewise.

${\begin{cases} 4 (6 - 2 a) + 8 \leq 24 & a \leq 3 \\ 4 (- (6 - 2 a)) + 8 \leq 24 & a > 3 \end{cases}$

Simplify $4 (6 - 2 a) + 8 \leq 24$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 4 \cdot 6 + 4 (- 2 a) + 8 \leq 24 & a \leq 3 \\ 4 (- (6 - 2 a)) + 8 \leq 24 & a > 3 \end{cases}$

Multiply $4$ by $6$ .

${\begin{cases} 24 + 4 (- 2 a) + 8 \leq 24 & a \leq 3 \\ 4 (- (6 - 2 a)) + 8 \leq 24 & a > 3 \end{cases}$

Multiply $- 2$ by $4$ .

${\begin{cases} 24 - 8 a + 8 \leq 24 & a \leq 3 \\ 4 (- (6 - 2 a)) + 8 \leq 24 & a > 3 \end{cases}$

Add $24$ and $8$ .

${\begin{cases} - 8 a + 32 \leq 24 & a \leq 3 \\ 4 (- (6 - 2 a)) + 8 \leq 24 & a > 3 \end{cases}$

Simplify $4 (- (6 - 2 a)) + 8 \leq 24$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} - 8 a + 32 \leq 24 & a \leq 3 \\ 4 (- 1 \cdot 6 - (- 2 a)) + 8 \leq 24 & a > 3 \end{cases}$

Multiply $- 1$ by $6$ .

${\begin{cases} - 8 a + 32 \leq 24 & a \leq 3 \\ 4 (- 6 - (- 2 a)) + 8 \leq 24 & a > 3 \end{cases}$

Multiply $- 2$ by $- 1$ .

${\begin{cases} - 8 a + 32 \leq 24 & a \leq 3 \\ 4 (- 6 + 2 a) + 8 \leq 24 & a > 3 \end{cases}$

Apply the distributive property.

${\begin{cases} - 8 a + 32 \leq 24 & a \leq 3 \\ 4 \cdot - 6 + 4 (2 a) + 8 \leq 24 & a > 3 \end{cases}$

Multiply $4$ by $- 6$ .

${\begin{cases} - 8 a + 32 \leq 24 & a \leq 3 \\ - 24 + 4 (2 a) + 8 \leq 24 & a > 3 \end{cases}$

Multiply $2$ by $4$ .

${\begin{cases} - 8 a + 32 \leq 24 & a \leq 3 \\ - 24 + 8 a + 8 \leq 24 & a > 3 \end{cases}$

Add $- 24$ and $8$ .

${\begin{cases} - 8 a + 32 \leq 24 & a \leq 3 \\ 8 a - 16 \leq 24 & a > 3 \end{cases}$

Solve $- 8 a + 32 \leq 24$ when $a \leq 3$ .

Solve $- 8 a + 32 \leq 24$ for $a$ .

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

Subtract $32$ from both sides of the inequality.

$- 8 a \leq 24 - 32$

Subtract $32$ from $24$ .

$- 8 a \leq - 8$

Divide each term in $- 8 a \leq - 8$ by $- 8$ and simplify.

Divide each term in $- 8 a \leq - 8$ by $- 8$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 8 a}{- 8} \geq \frac{- 8}{- 8}$

Simplify the left side.

Cancel the common factor of $- 8$ .

Cancel the common factor.

$\frac{- 8 a}{- 8} \geq \frac{- 8}{- 8}$

Divide $a$ by $1$ .

$a \geq \frac{- 8}{- 8}$

Simplify the right side.

Divide $- 8$ by $- 8$ .

$a \geq 1$

Find the intersection of $a \geq 1$ and $a \leq 3$ .

$1 \leq a \leq 3$

Solve $8 a - 16 \leq 24$ when $a > 3$ .

Solve $8 a - 16 \leq 24$ for $a$ .

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

Add $16$ to both sides of the inequality.

$8 a \leq 24 + 16$

Add $24$ and $16$ .

$8 a \leq 40$

Divide each term in $8 a \leq 40$ by $8$ and simplify.

Divide each term in $8 a \leq 40$ by $8$ .

$\frac{8 a}{8} \leq \frac{40}{8}$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 a}{8} \leq \frac{40}{8}$

Divide $a$ by $1$ .

$a \leq \frac{40}{8}$

Simplify the right side.

Divide $40$ by $8$ .

$a \leq 5$

Find the intersection of $a \leq 5$ and $a > 3$ .

$3 < a \leq 5$

Find the union of the solutions.

$1 \leq a \leq 5$

The result can be shown in multiple forms.

Inequality Form:

$1 \leq a \leq 5$

Interval Notation:

$[1, 5]$

Solve for a 4|6-2a|+8<=24

Related Questions

Support LCMGCF.com by disabling your adblocker.