In Exercises 59–94, solve each absolute value inequality. |2(x - 1) + 4| ≤ 8

Verified step by step guidance

Start by rewriting the absolute value inequality $ |2(x - 1) + 4| \leq 8 $ as a compound inequality without the absolute value: $ -8 \leq 2(x - 1) + 4 \leq 8 $.

Next, simplify the expression inside the inequality by distributing and combining like terms: $ 2(x - 1) + 4 = 2x - 2 + 4 = 2x + 2 $. So the inequality becomes $ -8 \leq 2x + 2 \leq 8 $.

Now, isolate the term with $ x $ by subtracting 2 from all parts of the inequality: $ -8 - 2 \leq 2x + 2 - 2 \leq 8 - 2 $, which simplifies to $ -10 \leq 2x \leq 6 $.

Then, solve for $ x $ by dividing all parts of the inequality by 2 (remember to keep the inequality signs the same since 2 is positive): $ \frac{-10}{2} \leq \frac{2x}{2} \leq \frac{6}{2} $, resulting in $ -5 \leq x \leq 3 $.

Finally, write the solution set as all $ x $ values between $ -5 $ and $ 3 $, inclusive, which can be expressed in interval notation as $ [-5, 3] $.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Absolute Value Inequalities

An absolute value inequality involves expressions within absolute value bars and compares them to a number. To solve |A| ≤ B, where B ≥ 0, rewrite it as a compound inequality: -B ≤ A ≤ B. This approach helps find the range of values for the variable that satisfy the inequality.

Distributive Property

The distributive property allows you to multiply a single term across terms inside parentheses: a(b + c) = ab + ac. Applying this property simplifies expressions like 2(x - 1) into 2x - 2, making it easier to isolate variables and solve inequalities.

Solving Linear Inequalities

Solving linear inequalities involves isolating the variable on one side using inverse operations while maintaining inequality direction. When multiplying or dividing by a negative number, the inequality sign reverses. This process helps determine the solution set for the variable.

Recommended video:

06:07

Linear Inequalities

Related Practice

Textbook Question

Solve each equation in Exercises 65–74 using the quadratic formula. $x^{2} + 8 x + 15 = 0$

Textbook Question

Perform the indicated operation(s) and write the result in standard form. (2 - 3i)(1 - i) - (3 - i)(3 + i)

views

Textbook Question

Solve each equation in Exercises 47–64 by completing the square. 3x² - 5x - 10 = 0

Textbook Question

In Exercises 59–94, solve each absolute value inequality. |3x + 5| < 17

Textbook Question

In Exercises 61–76, solve each absolute value equation or indicate that the equation has no solution. |2x - 1| = 5

views

Textbook Question

Determine the constant that should be added to the binomial so that it becomes a perfect square trinomial. Then write and factor the trinomial. $x squared plus 20 x$