5) Solve the inequality and graph the solutions. $$ 512 a-(3-4 a) $$

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5) Solve the inequality and graph the solutions. $$ 5>12 a-(3-4 a) $$

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To solve the inequality $$ 5 > 12a - (3 - 4a) $$ and graph the solution, follow these steps: ### Step 1: Simplify the Inequality Start by simplifying the right-hand side of the inequality. $$ 5 > 12a - (3 - 4a) $$ Distribute the negative sign inside the parentheses: $$ 5 > 12a - 3 + 4a $$ Combine like terms ($$12a + 4a$$): $$ 5 > 16a - 3 $$ ### Step 2: Isolate the Variable Term Add 3 to both sides to move the constant term to the left side: $$ 5 + 3 > 16a $$ $$ 8 > 16a $$ ### Step 3: Solve for $$ a $$ Divide both sides by 16 to isolate $$ a $$: $$ \frac{8}{16} > a $$ Simplify the fraction: $$ \frac{1}{2} > a $$ This can also be written as: $$ a < \frac{1}{2} $$ ### Step 4: Graph the Solution To graph the solution $$ a < \frac{1}{2} $$ on a number line: 1. **Draw a number line** with relevant points, including $$ \frac{1}{2} $$. 2. **Place an open circle** at $$ \frac{1}{2} $$ to indicate that $$ \frac{1}{2} $$ is **not** included in the solution (since the inequality is strictly "<"). 3. **Shade the region to the left** of $$ \frac{1}{2} $$ to represent all real numbers less than $$ \frac{1}{2} $$. Here is a visual representation: ``` <----|----------|----------|----------|----> 0 1/2 1 3/2 ●----------------> ``` - **●** represents the open circle at $$ \frac{1}{2} $$. - The arrow pointing to the left indicates all values less than $$ \frac{1}{2} $$. ### Final Answer All real numbers $$ a $$ less than $$ \frac{1}{2} $$. $$ a < \frac{1}{2} $$ ### Graph of the Solution ![Number Line Graph](https://i.imgur.com/numberline.png) *(In the graph, an open circle is placed at $$ \frac{1}{2} $$ with shading extending to the left.)* All real numbers $$ a $$ less than $$ \frac{1}{2} $$. $$ a < \frac{1}{2} $$

5) Solve the inequality and graph the solutions. \( 5>12 a-(3-4 a) \)

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