6-42 Solve each of the following inequalities Represent the solutions algebraically (with symbols) and graphically (on a number line)Homework Help 3x-3lt 2-2x b. (4)/(5)xgeqslant 8

Let's solve each inequality step by step.<br /><br />### Inequality 1: \(3x - 3 < 2 - 2x\)<br /><br />1. **Combine like terms:**<br /> \[<br /> 3x - 3 < 2 - 2x<br /> \]<br /> Add \(2x\) to both sides:<br /> \[<br /> 3x + 2x - 3 < 2<br /> \]<br /> Simplify:<br /> \[<br /> 5x - 3 < 2<br /> \]<br /><br />2. **Isolate the variable term:**<br /> Add 3 to both sides:<br /> \[<br /> 5x - 3 + 3 < 2 + 3<br /> \]<br /> Simplify:<br /> \[<br /> 5x < 5<br /> \]<br /><br />3. **Solve for \(x\):**<br /> Divide both sides by 5:<br /> \[<br /> x < 1<br /> \]<br /><br />**Algebraic Solution:**<br />\[<br />x < 1<br />\]<br /><br />**Graphical Solution:**<br />On a number line, represent all values less than 1 with an open circle at 1 and shade to the left.<br /><br />```<br /><---|---|---|---|---|---|---|---|---|---><br /> -2 -1 0 1 2 3 4 5 6<br /> o================><br />```<br /><br />### Inequality 2: \(\frac{4}{5}x \geq 8\)<br /><br />1. **Isolate \(x\):**<br /> Multiply both sides by the reciprocal of \(\frac{4}{5}\), which is \(\frac{5}{4}\):<br /> \[<br /> x \geq 8 \times \frac{5}{4}<br /> \]<br /> Simplify:<br /> \[<br /> x \geq 10<br /> \]<br /><br />**Algebraic Solution:**<br />\[<br />x \geq 10<br />\]<br /><br />**Graphical Solution:**<br />On a number line, represent all values greater than or equal to 10 with a closed circle at 10 and shade to the right.<br /><br />```<br /><---|---|---|---|---|---|---|---|---|---><br /> -2 -1 0 1 2 3 4 5 6<br /> •================><br />```<br /><br />### Summary<br /><br />- For \(3x - 3 < 2 - 2x\), the solution is \(x < 1\).<br />- For \(\frac{4}{5}x \geq 8\), the solution is \(x \geq 10\).