Problem 102 - Algebra and Trigonometry

Question

In Exercises 95–102, use interval notation to represent all values of x satisfying the given conditions. \(y=8-|5 x+3|\) and \(y\) is at least 6

Step-by-Step Solution

Answer

The interval representation of all values of \(x\) that satisfy the given condition is \([-1, -0.2]\)

Step 1: Set up the inequality

First set up the inequality based on the condition given. The condition is \(y\) is at least 6, so the inequality that needs to be solved is: \(8-|5x+3| \geq 6\)

Step 2: Simplify the inequality

Next, simplify the inequality by subtracting 8 from both sides: \(-|5x+3| \geq -2\) Or rephrase it: \(|5x+3| \leq 2\)

Step 3: Solve the inequality

Next, we need to find the interval that satisfy the absolute value inequality. An absolute value inequality of the form \(|a| \leq b\) translates to \(-b \leq a \leq b\). So, our inequality becomes \(-2 \leq 5x+3 \leq 2\) This inequality can be split into two: \(-2 \leq 5x+3\) and \(5x+3 \leq 2\)

Step 4: Solve the two inequalities

Solving \(-2 \leq 5x+3\): Subtract 3 to obtain \(-5 \leq 5x\), then divide by 5: \(-1 \leq x\). Solving \(5x+3 \leq 2\): Subtract 3 to obtain \(5x \leq -1\), then divide by 5: \(x \leq -0.2\)

Step 5: Combine the Solutions

The solution to the original inequality is the interval of \(x\) which satisfies both inequalities. Here, \(x\) is greater than or equal to -1 and less than or equal to -0.2, so the solution set is the intersection of these two solutions and the result is \([-1, -0.2]\)

Key Concepts

Understanding Absolute Value Inequalities

An absolute value inequality involves finding the range of values that satisfy an inequality containing an absolute value. The absolute value \(|a|\) represents the distance of \(a\) from zero on a number line. It is always non-negative. In problems like \(|5x+3| \leq 2\), we explore when the expression within the absolute value represents numbers between -2 and 2.

To break it down, \(|a| \leq b\) translates to \(-b \leq a \leq b\). This kind of setup results in two separate inequalities that you solve individually. This allows us to capture both the positive and negative scenarios of the expression within the absolute value. It's why you see absolute value inequalities opening you up to multiple potential solutions in terms of intervals.

Solving the Inequality Solution

After establishing the absolute value inequality, the next challenge is to solve it step by step. Here, we simplify first by aligning our expression to a straightforward form. This might involve operations like subtraction or adding terms on both sides. For instance, reducing \(8 - |5x+3| \geq 6\) to \(|5x+3| \leq 2\) helps us focus purely on the inequality aspect.

Breaking it down further into simpler inequalities, like \(-2 \leq 5x+3\) and \(5x+3 \leq 2\), brings clarity. You're presented with two straightforward inequality problems: one on the left and another on the right of the inequality sign. Solve them independently, and the intersection of solutions gives the final interval. This controlled approach makes sure each step remains logical and connected, minimizing confusion.

Understanding Algebraic Expressions

Algebraic expressions form the basis of equations and inequalities. They consist of numbers, variables, and operation signs. For absolute value inequalities, expressions like \(5x+3\) need keen manipulation. Knowing how to handle them—by rearranging or isolating variables—is crucial.

In our problem's context, the expression inside the absolute value symbols serves as our main focus. Whatever alterations we make to this expression redefine the range of solutions. For instance, solving \(5x+3\leq 2\) means isolating \(x\). By systematically subtracting or dividing through algebraic manipulation, you can transform and solve such expressions efficiently. Grasping such skills in handling algebraic expressions is vital for successfully navigating through various math challenges.