Question Solve the absolute value inequality $|4x+6| < 9$ to find the range of values for $x$ .

Studdy Solution

STEP 1

1. The expression $|4x+6|<9$ represents an inequality involving an absolute value.
2. The absolute value of a number is the distance of that number from zero on the real number line, and it is always non-negative.
3. To solve an absolute value inequality such as $|A|<B$ , where $A$ is an expression and $B$ is a positive number, we can split it into two separate inequalities: $-B<A<B$ .

STEP 2

1. Split the absolute value inequality into two separate linear inequalities.
2. Solve each linear inequality for $x$ .
3. Combine the solutions to form the final solution set.

STEP 3

Split the absolute value inequality $|4x+6|<9$ into two separate inequalities: $-9<4x+6<9$ .

STEP 4

Write the two separate inequalities explicitly:
$-9 < 4x + 6$ $4x + 6 < 9$

STEP 5

Solve the first inequality $-9 < 4x + 6$ for $x$ .
$-9 < 4x + 6$

STEP 6

Subtract 6 from both sides of the first inequality to isolate the term with $x$ .
$-9 - 6 < 4x + 6 - 6$

STEP 7

Simplify the first inequality after subtraction.
$-15 < 4x$

STEP 8

Divide both sides of the first inequality by 4 to solve for $x$ .
$\frac{-15}{4} < x$

STEP 9

Write the first part of the solution set from the first inequality.
$x > \frac{-15}{4}$

STEP 10

Solve the second inequality $4x + 6 < 9$ for $x$ .
$4x + 6 < 9$

STEP 11

Subtract 6 from both sides of the second inequality to isolate the term with $x$ .
$4x + 6 - 6 < 9 - 6$

STEP 12

Simplify the second inequality after subtraction.
$4x < 3$

STEP 13

Divide both sides of the second inequality by 4 to solve for $x$ .
$x < \frac{3}{4}$

STEP 14

Write the second part of the solution set from the second inequality.
$x < \frac{3}{4}$

STEP 15

Combine the solutions from the two inequalities to form the final solution set.
$\frac{-15}{4} < x < \frac{3}{4}$
The solution to the inequality $|4x+6|<9$ is:
$\frac{-15}{4} < x < \frac{3}{4}$

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