Solve and graph the following absolute value inequalities. ( B ) ∣ 6 x − 9 ∣ − 7 ≤ − 7 \left|6x-9\right|-7\le-7

Welcome back. For this problem, we'd like to solve the inequality absolute value of six x minus nine minus seven is less than or equal to seven. The first step is going to be isolating the absolute value on the left side of the inequality so that we can rewrite this inequality as a three part inequality to solve. So we need to get rid of this minus seven on the left, which we can do by adding seven to both sides of the inequality. Canceling the minus seven on the left, leaving us with absolute value of six x minus nine, is less than or equal to negative seven plus seven, which is zero. Now as a reminder, distance can't be negative, which means distance can't be less than zero. And because absolute value gives us a distance, we know that absolute value also can't be less than zero. So if the absolute value of six x minus nine is less than or equal to zero, but it can't actually be less than zero, that means it must be equal to zero. So we have that the absolute value of six x minus nine is equal to zero. As we saw when going over absolute value equations, this means that the thing inside of the absolute value, six x minus nine, must be equal to zero. Now we can solve this equation to get the one solution to our inequality. We can start by adding nine to both sides of the equation to cancel the minus nine on the left, leaving us with six x equals nine, and then divide both sides of the equation by six to get x equals nine sixths. Then we can simplify nine sixths to three halves, and three halves is just 1.5. So the single solution to our inequality is x equals 1.5. To graph this solution, we just put a single dot on our number line at 1.5 to represent again that one solution to our inequality, x equals 1.5. Great work.

Solve and graph the following absolute value inequalities. ( B ) ∣ 6 x − 9 ∣ − 7 ≤ − 7 \left|6x-9\right|-7\le-7

[ − 1.5 , 1.5 ] \left\lbrack-1.5,1.5\right\rbrack [ − 1.5 , 1.5 ]

( − 1.5 , 1.5 ) \left(-1.5,1.5\right) ( − 1.5 , 1.5 )

2. Linear Equations and Inequalities

Absolute Value Inequalities

Intermediate Algebra

2. Linear Equations and Inequalities

Absolute Value Inequalities

Struggling with Intermediate Algebra?

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Multiple Choice

Solve and graph the following absolute value inequalities.
(B) $∣ 6 x - 9 ∣ - 7 \leq - 7$

$x equals 1.5$

Number line from -8 to 8 with a red point marked at 1.5 on the x-axis.

$x = - 1.5$

Number line from -8 to 8 with a red point marked at -2 representing a solution to an absolute value inequality.

$\left\lbrack-1.5,1.5\right\rbrack$

Number line from -8 to 8 with the interval between -1 and 2 highlighted in red brackets.

$\left(-1.5,1.5\right)$

Number line graph showing the solution to the absolute value inequality |6x - 9| - 7 ≤ -7 with highlighted interval.

Verified step by step guidance

Start with the given inequality: $\left|6x - 9\right| - 7 \leq -7$.

Add 7 to both sides to isolate the absolute value expression: $\left|6x - 9\right| - 7 + 7 \leq -7 + 7$, which simplifies to $\left|6x - 9\right| \leq 0$.

Recall that the absolute value of any expression is always greater than or equal to zero, so $\left|6x - 9\right| \leq 0$ means $\left|6x - 9\right| = 0$.

Set the expression inside the absolute value equal to zero: \$6x - 9 = 0$.

Solve for $x$ by adding 9 to both sides and then dividing by 6: \$6x = 9$ so $x = \frac{9}{6}$, which simplifies to $x = \frac{3}{2}$.

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Struggling with Intermediate Algebra?

Solve and graph the following absolute value inequalities.(B) ∣6x−9∣−7≤−7\left|6x-9\right|-7\le-7

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Solve and graph the following absolute value inequalities.
(B) $∣ 6 x - 9 ∣ - 7 \leq - 7$