Absolute Value Inequalities

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Algebra › Absolute Value Inequalities

Questions 1 - 10

Solve this inequality.

$\left | \frac{4x+3}{x+6}\right |\eq>1$

$x \eq< -\frac{9}{5}, \hspace x \eq>1$

$x\eq>-\frac{9}{5}, x \eq<1$

$x\eq>-\frac{9}{5}, x\eq>1$

$x<-\frac{9}{5}, x\eq<1$

$x\eq<\frac{9}{5}, x\eq>-1$

Explanation

Split the inequality into two possible cases as follows, based on the absolute values.

First case: $\frac{4x+3}{x+6}\eq>1$

Second case: $\frac{4x+3}{x+6} \eq< -1$

Let's find the inequality of the first case.

$\frac{4x+3}{x+6} \eq>1$

Multiply both sides by x + 6.

$4x+3 \eq>x+6$

Subtract x from both sides, then subtract 3 from both sides.

$3x \eq>3$

Divide both sides by 3.

$x\eq>1$

Let's find the inequality of the second case.

$\frac{4x+3}{x+6} \eq< -1$

Multiply both sides by x + 6.

$4x+3 \eq< -1(x+6)$

Simplify.

Add x to both sides, then subtract 3 from both sides.

$5x\eq< -9$

Divide both sides by 5.

$x \eq<-\frac{9}{5}$

So the range of x-values is $x<-\frac{9}{5}$ and $x \eq>1$ .

Solve for :

The inequality has no solution.

$\left ( -5, 4 \frac{1}{2} \right )$

$\left ( - 4 \frac{1}{2}, 5 \right )$

$\left ( -5, -4 \frac{1}{2} \right )$

$\left ( 4 \frac{1}{2}, 5 \right )$

Explanation

The absolute value of a number must always be nonnegative, so can never be less than . This means the inequality has no solution.

Solve the inequality:

$\small \left |15x - 39\right | < -42$

$\small \varnothing$ (no solution)

$\small \left(-\frac{1}{5}, \frac{61}{5} \right )$

$\small \left ( -\infty , -\frac{1}{5}\right ) \cup \left (\frac{27}{5} , \infty \right )$

$\small \left ( -\infty, \infty \right )$

Explanation

The inequality compares an absolute value function with a negative integer. Since the absolute value of any real number is greater than or equal to 0, it can never be less than a negative number. Therefore, $\small \left |15x -39 \right| < -42$ can never happen. There is no solution.

Give the solution set for the following equation:

$\begin{Bmatrix} -18.5, 22 \end{Bmatrix}$

$\begin{Bmatrix} 22 \end{Bmatrix}$

$\begin{Bmatrix} -24.5, 22 \end{Bmatrix}$

$\begin{Bmatrix} 22, 24.5 \end{Bmatrix}$

$\begin{Bmatrix} 18.5, 22 \end{Bmatrix}$

Explanation

First, subtract 5 from both sides to get the absolute value expression alone.

Split this into two linear equations:

$x = -74 \div 4 = -18.5$

The solution set is $\begin{Bmatrix} -18.5, 22 \end{Bmatrix}$

Solve: $3\left | 3x+3\right |<36$

$(-\infty,-5)\bigcup (3, \infty)$

$(-\infty,-5]\bigcup [3, \infty)$

$(-\infty,-5)\bigcup (-5,3)\bigcup (3, \infty)$

Explanation

Solve by first dividing three on both sides. We will need to isolate the absolute value sign before splitting the absolute value into its positive and negative components.

$\frac{3\left | 3x+3\right |}{3}<\frac{36}{3}$

$\left | 3x+3\right |<12$

Split the absolute value into two equations.

Solve the first equation. Subtract three on both sides and simplify.

Divide by three on both sides.

This is the first solution. Solve the second equation. Divide by a negative one on both sides. This will switch the inequality sign.

$\frac{-(3x+3)}{-1}<\frac{12}{-1}$

Subtract three from both sides and simplify.

Divide by three on both sides.

The answer is between negative five and three but does not include both numbers:

Solve the inequality for .

$\left | 3x+5\right |\geq 1$

$x\leq -2$ and $x\geq -\frac{4}{3}$

$x\geq -2$

$x\geq -\frac{4}{3}$ and $x\leq \frac{4}{3}$

$x\geq -\frac{4}{3}$

$-2\leq x\leq -\frac{4}{3}$

Explanation

To solve an inequality with absolute value you have to consider the two equations it creates.

$\left | 3x+5\right |\geq 1$ becomes $3x+5\geq 1$ and $3x+5\leq -1$

Solve for both inequalities by following the balancing rules. Be careful of division or multiplication of a negative number; if that happens, flip the inequality sign.

$3x+5\geq 1$

$3x\geq -4$

$x\geq -\frac{4}{3}$

$3x+5\leq -1$

$3x\leq -6$

$x\leq -2$

Solve the inequality $\left | 4x-1\right |+7\leq14$ .

$-1.5\leq x\leq2$

$-6\leq x\leq 0.5$

$-2\leq x\leq 1.5$

$x\leq1$

$x\geq-3$

Explanation

First, we can simplify this inequality by subtracting 7 from both sides. This gives us

$\left | 4x-1\right |\leq7$

Next, however, we need to make two separate inequalities due to the presence of an absolute value expression. What this inequality actually means is that

$4x-1\leq7$

and

$4x-1\geq -7$

(Be careful with the inequality signs here! The second sign must be switched to allow for the effect of absolute value on negative numbers. In other words, the inequality must be greater than because, after the absolute value is applied, it will be less than 7.) When we solve the two inequalities, we get two solutions:

$x\leq2$

and

$x\geq-1.5$

For the original statement to be true, both of these inequalities must be fulfilled. We're left with a final answer of

$-1.5\leq x\leq2$

$\textup{If }\left | 2x+4\right |\leq10\textup{ , what are all possible values of }x\textup{?}$

$-7\leq x\leq3$

$-3\leq x\leq3$

$x\leq-7\textup{ and }x\geq3$

$-3\leq x\leq7$

Explanation

$\textup{Set up two inequalities. Reverse the inequality sign for the negative one.}$

$2x+4\leq10;;;;;;;;;;;;;2x+4\geq-10$

$2x\leq6;;;;;;;;;;;;;;;;;; ; 2x\geq-14$

$x\leq3;;;;;;;;;;;;;;;;;;;;;;;x\geq-7$

Solve the equation in interval notation: $-3\left | x\right |>6$

$\textup{No Solution.}$

$(-\infty, -2)\bigcup(2, \infty)$

$(-\infty, -2]\bigcup[2, \infty)$

Explanation

Divide by negative three on both sides. Dividing by a negative number will switch the sign.

$\frac{-3\left | x\right |}{-3}>\frac{6}{-3}$

Simplify the fractions and switch the sign.

$\left | x\right | <-2$

We will notice that this has no solution because absolute value converts all values to a positive number. There are no such values inside an absolute value that will be less than negative two.

The answer is: $\textup{No Solution.}$

Solve the inequality in interval notation: $\left | 3x+7\right |>-2x-3$

$(-\infty,-4)\bigcup(-2,\infty)$

$(-\infty,-4)\bigcup(2,\infty)$

$(-\infty,-4)\bigcup(4,\infty)$

$(-\infty,-2)\bigcup(4,\infty)$

$\textup{There is no solution.}$

Explanation

Split the absolute value into its positive and negative inequalities. Be sure to encapsulate the left quantity with a negative sign for the second inequality.

First inequality: