Problem 45

Question

Solve the equation (a) graphically, (b) numerically, and (c) symbolically. Then solve the nelated inequality. $$|2 x-5|=10, \quad|2 x-5|<10$$

Step-by-Step Solution

Verified

Answer

The solutions to the equation are $x=7.5$ and $x=-2.5$. The solution to the inequality is \(-2.5

1Step 1: Understanding the Absolute Value Equation

The given equation is $|2x-5|=10$. The absolute value $|a|$ represents the distance of $a$ from zero on a number line, resulting in two scenarios: $2x-5=10$ and $2x-5=-10$. We will solve each part separately.

2Step 2: Solving the Equation Graphically

Graph the equations $y=|2x-5|$ and $y=10$ on the same axes. The solution to $|2x-5|=10$ is where these two lines intersect. Find the $x$ values at these intersection points.

3Step 3: Solving the Equation Numerically

For the equation $|2x-5|=10$, set up the two equal equations: $2x-5=10$ and $2x-5=-10$. Solve for $x$ in each case. For $2x-5=10$, we have $2x = 15$ leading to $x=7.5$. For $2x-5=-10$, we have $2x = -5$ leading to $x=-2.5$. Thus, the solutions are $x=7.5$ and $x=-2.5$.

4Step 4: Solving the Equation Symbolically

Start with the equation $|2x-5|=10$. Rewrite it as the two equations: $2x-5=10$ or $2x-5=-10$. Solving $2x-5=10$ gives $x=7.5$. Solving $2x-5=-10$ gives $x=-2.5$. Therefore, the solutions for $x$ are $x=7.5$ and $x=-2.5$.

5Step 5: Solving the Inequality Graphically

Graph $y = |2x - 5|$ and $y = 10$. For $|2x-5|<10$, identify the range of $x$-values where $y = |2x-5|$ is below the line $y = 10$. This is between the two intersection points calculated earlier.

6Step 6: Solving the Inequality Numerically

For $|2x-5|<10$, solve the two inequalities: $2x-5<10$ and $2x-5>-10$. For $2x-5<10$, add 5 to get $2x<15$, then divide by 2 to find $x<7.5$. For $2x-5>-10$, add 5 to get $2x>-5$, then divide by 2 to find $x>-2.5$. Combining these gives \(-2.5

7Step 7: Solving the Inequality Symbolically

Write $|2x-5|<10$ as two inequalities: $2x-5<10$ and $2x-5>-10$. Solving $2x-5<10$ gives $x<7.5$ and solving $2x-5>-10$ gives $x>-2.5$. Therefore, the solution to the inequality is \(-2.5

Key Concepts

Absolute Value EquationsInequalitiesGraphical SolutionNumerical SolutionSymbolic Solution

Absolute Value Equations

Absolute value equations such as $|2x-5|=10$ involve understanding the meaning of the absolute value symbol, $|a|$. Absolute value refers to the distance of a number from zero on the number line, which is always non-negative. This creates two scenarios:

$2x-5=10$ leading to straightforward calculations
$2x-5=-10$ considering the equivalent negative possibility

By solving each scenario separately, you can find the values of $x$ that satisfy the equation.

Inequalities

Inequalities are similar to equations but they involve comparisons, like $|2x-5|<10$. This means we are looking for the set of $x$ values where the expression inside the absolute value is less than 10.

Transform $|2x-5|<10$ into two separate inequalities: $2x-5<10$ and $2x-5>-10$
Solve each inequality individually: $x<7.5$ and $x>-2.5$
Combine the solutions to get a range: $-2.5 < x < 7.5$

This range represents all the possible values of $x$ that make the inequality true.

Graphical Solution

A graphical solution involves drawing out an equation or inequality on a coordinate system. For the equation $|2x-5|=10$, you would graph:

The absolute value function: $y=|2x-5|$
A horizontal line: $y=10$

The solution to the equation is where these two graphs intersect. Identifying these points visually allows us to find the solutions, in this case, $x=7.5$ and $x=-2.5$. For the inequality, observe where the graph of $y=|2x-5|$ is below $y=10$, which is the interval $-2.5 < x < 7.5$. Graphing provides a visual verification of calculated solutions.

Numerical Solution

Numerical solutions involve solving the problems using basic arithmetic procedures. For the equation $|2x-5|=10$, convert it into two separate equations: