4) Solve the inequality. Then explain how the solution to an inequality is different than an equation. \( 10-2(3 x+4)<11 \)

To solve the inequality and understand how its solution differs from that of an equation, let's proceed step-by-step. ### **Solving the Inequality** Given the inequality: \[ 10 - 2(3x + 4) < 11 \] 1. **Expand the Expression:** \[ 10 - 2 \cdot 3x - 2 \cdot 4 < 11 \] \[ 10 - 6x - 8 < 11 \] 2. **Combine Like Terms:** \[ (10 - 8) - 6x < 11 \] \[ 2 - 6x < 11 \] 3. **Isolate the Variable Term:** Subtract 2 from both sides: \[ 2 - 6x - 2 < 11 - 2 \] \[ -6x < 9 \] 4. **Solve for \( x \):** Divide both sides by -6. **Remember:** When you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign. \[ \frac{-6x}{-6} > \frac{9}{-6} \] \[ x > -\frac{3}{2} \] ### **Solution** \[ x > -\frac{3}{2} \] This means that any real number greater than \(-\frac{3}{2}\) satisfies the inequality. ### **Difference Between Solutions to Inequalities and Equations** 1. **Nature of Solutions:** - **Equations:** Typically have specific solutions. For example, \(2x + 3 = 7\) has the unique solution \(x = 2\). - **Inequalities:** Represent a range or interval of solutions. In our case, \(x > -\frac{3}{2}\) includes infinitely many values greater than \(-\frac{3}{2}\). 2. **Graphical Representation:** - **Equations:** Solutions are points on a number line. - **Inequalities:** Solutions are intervals on a number line, often represented with open or closed circles and arrows indicating the direction of the inequality. 3. **Solution Process:** - **Equations:** Operations performed are the same on both sides without changing the direction of equality. - **Inequalities:** Special rules apply, such as reversing the inequality sign when multiplying or dividing by a negative number. 4. **Applications:** - **Equations:** Used to find exact values, such as determining when two expressions are equal. - **Inequalities:** Used to describe scenarios with constraints, such as determining the range of possible values that satisfy certain conditions. **In summary**, while equations seek specific values that make both sides equal, inequalities describe a set of possible values that satisfy a particular condition or range.