Basic Math Examples

$| 4 x - 10 | \geq 3$

Step 1

Write $| 4 x - 10 | \geq 3$ as a piecewise.

Tap for more steps...

Step 1.1

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

$4 x - 10 \geq 0$

Step 1.2

Solve the inequality.

Tap for more steps...

Step 1.2.1

Add $10$ to both sides of the inequality.

$4 x \geq 10$

Step 1.2.2

Divide each term in $4 x \geq 10$ by $4$ and simplify.

Tap for more steps...

Step 1.2.2.1

Divide each term in $4 x \geq 10$ by $4$ .

$\frac{4 x}{4} \geq \frac{10}{4}$

Step 1.2.2.2

Simplify the left side.

Tap for more steps...

Step 1.2.2.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} \geq \frac{10}{4}$

Step 1.2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{10}{4}$

Step 1.2.2.3

Simplify the right side.

Tap for more steps...

Step 1.2.2.3.1

Cancel the common factor of $10$ and $4$ .

Tap for more steps...

Step 1.2.2.3.1.1

Factor $2$ out of $10$ .

$x \geq \frac{2 (5)}{4}$

Step 1.2.2.3.1.2

Cancel the common factors.

Tap for more steps...

Step 1.2.2.3.1.2.1

Factor $2$ out of $4$ .

$x \geq \frac{2 \cdot 5}{2 \cdot 2}$

Step 1.2.2.3.1.2.2

Cancel the common factor.

$x \geq \frac{2 \cdot 5}{2 \cdot 2}$

Step 1.2.2.3.1.2.3

Rewrite the expression.

$x \geq \frac{5}{2}$

Step 1.3

In the piece where $4 x - 10$ is non-negative, remove the absolute value.

$4 x - 10 \geq 3$

Step 1.4

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

$4 x - 10 < 0$

Step 1.5

Solve the inequality.

Tap for more steps...

Step 1.5.1

Add $10$ to both sides of the inequality.

$4 x < 10$

Step 1.5.2

Divide each term in $4 x < 10$ by $4$ and simplify.

Tap for more steps...

Step 1.5.2.1

Divide each term in $4 x < 10$ by $4$ .

$\frac{4 x}{4} < \frac{10}{4}$

Step 1.5.2.2

Simplify the left side.

Tap for more steps...

Step 1.5.2.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 1.5.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} < \frac{10}{4}$

Step 1.5.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{10}{4}$

Step 1.5.2.3

Simplify the right side.

Tap for more steps...

Step 1.5.2.3.1

Cancel the common factor of $10$ and $4$ .

Tap for more steps...

Step 1.5.2.3.1.1

Factor $2$ out of $10$ .

$x < \frac{2 (5)}{4}$

Step 1.5.2.3.1.2

Cancel the common factors.

Tap for more steps...

Step 1.5.2.3.1.2.1

Factor $2$ out of $4$ .

$x < \frac{2 \cdot 5}{2 \cdot 2}$

Step 1.5.2.3.1.2.2

Cancel the common factor.

$x < \frac{2 \cdot 5}{2 \cdot 2}$

Step 1.5.2.3.1.2.3

Rewrite the expression.

$x < \frac{5}{2}$

Step 1.6

In the piece where $4 x - 10$ is negative, remove the absolute value and multiply by $- 1$ .

$- (4 x - 10) \geq 3$

Step 1.7

Write as a piecewise.

${\begin{cases} 4 x - 10 \geq 3 & x \geq \frac{5}{2} \\ - (4 x - 10) \geq 3 & x < \frac{5}{2} \end{cases}$

Step 1.8

Simplify $- (4 x - 10) \geq 3$ .

Tap for more steps...

Step 1.8.1

Apply the distributive property.

${\begin{cases} 4 x - 10 \geq 3 & x \geq \frac{5}{2} \\ - (4 x) - - 10 \geq 3 & x < \frac{5}{2} \end{cases}$

Step 1.8.2

Multiply $4$ by $- 1$ .

${\begin{cases} 4 x - 10 \geq 3 & x \geq \frac{5}{2} \\ - 4 x - - 10 \geq 3 & x < \frac{5}{2} \end{cases}$

Step 1.8.3

Multiply $- 1$ by $- 10$ .

${\begin{cases} 4 x - 10 \geq 3 & x \geq \frac{5}{2} \\ - 4 x + 10 \geq 3 & x < \frac{5}{2} \end{cases}$

Step 2

Solve $4 x - 10 \geq 3$ for $x$ .

Tap for more steps...

Step 2.1

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

Tap for more steps...

Step 2.1.1

Add $10$ to both sides of the inequality.

$4 x \geq 3 + 10$

Step 2.1.2

Add $3$ and $10$ .

$4 x \geq 13$

Step 2.2

Divide each term in $4 x \geq 13$ by $4$ and simplify.

Tap for more steps...

Step 2.2.1

Divide each term in $4 x \geq 13$ by $4$ .

$\frac{4 x}{4} \geq \frac{13}{4}$

Step 2.2.2

Simplify the left side.

Tap for more steps...

Step 2.2.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} \geq \frac{13}{4}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{13}{4}$

Step 3

Solve $- 4 x + 10 \geq 3$ for $x$ .

Tap for more steps...

Step 3.1

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

Tap for more steps...

Step 3.1.1

Subtract $10$ from both sides of the inequality.

$- 4 x \geq 3 - 10$

Step 3.1.2

Subtract $10$ from $3$ .

$- 4 x \geq - 7$

Step 3.2

Divide each term in $- 4 x \geq - 7$ by $- 4$ and simplify.

Tap for more steps...

Step 3.2.1

Divide each term in $- 4 x \geq - 7$ by $- 4$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 4 x}{- 4} \leq \frac{- 7}{- 4}$

Step 3.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.2.1

Cancel the common factor of $- 4$ .

Tap for more steps...

Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} \leq \frac{- 7}{- 4}$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{- 7}{- 4}$

Step 3.2.3

Simplify the right side.

Tap for more steps...

Step 3.2.3.1

Dividing two negative values results in a positive value.

$x \leq \frac{7}{4}$

Step 4

Find the union of the solutions.

$x \leq \frac{7}{4}$ or $x \geq \frac{13}{4}$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x \leq \frac{7}{4} or x \geq \frac{13}{4}$

Interval Notation:

$(- \infty, \frac{7}{4}] \cup [\frac{13}{4}, \infty)$

Step 6