Step 1: Use the distributive property, if needed.

Distribute on the left-hand side of the inequality.

$2\left( x + \frac{1}{2} \right) \gt x$

$2x + 1 \gt x$

Step 2: Collect the like terms on each side of the inequality, if needed.

There are no like terms to combine on either side.

Step 3: Use the properties of inequality to collect all the variable terms on one side of the inequality and all the constant terms on the other side.

$2x + 1 \gt x$

$2x \color{#A80000}{- 2x} + 1 \gt x \color{#A80000}{- 2x}$

$1 \gt - x$

Step 4: Use inverse operations and the properties of inequality to finish solving.

Apply the division property of inequality. Remember to reverse the inequality symbol.

$\frac{1}{- 1} \gt - \frac{x}{- 1}$

$- 1 \lt x$

Step 5: Graph your answer on a number line.

Step 6: Check your shading.

Choose any value in the shaded area.

Substitute $x = 0.$

$2\left( \left( 0 \right) + \frac{1}{2} \right) \gt \left( 0 \right)$

$2\left( \frac{1}{2} \right) \gt 0$

$1 \gt 0$

Step 1: Use the distributive property, if needed.

$30 + 16x \geq 12\left( 2x + 15 \right)$

$30 + 16x \geq 24x + 180$

Step 2: Collect the like terms on each side of the inequality, if needed.

There are no like terms to combine on either side.

Step 3: Use the properties of inequality to collect all the variable terms on one side of the inequality and all the constant terms on the other side.

Collect the variable terms on the right-hand side of the inequality to keep the coefficient positive.

$30 + 16x \geq 24x + 30$

$30 + 16x \color{#A80000}{- 16x} \geq 24x \color{#A80000}{- 16x} + 180$

$30 \geq 8x + 180$

$30 \color{#A80000}{- 180} \geq 8x + 180 \color{#A80000}{- 180}$

$- 150 \geq 8x$

Step 4: Use inverse operations and the properties of inequality to finish solving.

Apply the division property of inequality.

$- \frac{150}{\color{#A80000}{8}} \geq \frac{8x}{\color{#A80000}{8}}$

$- 18.75 \geq x$

Step 5: Graph your answer on a number line.

Step 6: Check your shading.

Choose any value in the shaded area.

Substitute $x = - 20$

$30 + 16( - 20) \geq 12\left( 2( - 20) + 15 \right)$

$30 + 16\left( - 20 \right) \geq 12\left( - 40 + 15 \right)$

$30 + 16\left( - 20 \right) \geq 12\left( - 25 \right)$

$30 - 320 \geq - 300$

$- 290 \geq - 300$

Step 1: Use the distributive property, if needed.

$2x + 4\left( x - 42.5 \right) \geq 9x - 425$

$2x + 4x - 170 \geq 9x - 425$

Step 2: Collect the like terms on each side of the inequality, if needed.

$\color{#A80000}{2x + 4x} - 170 \geq 9x - 425$

$\color{#A80000}{6x} - 170 \geq 9x - 425$

Step 3: Use the properties of inequality to collect all the variable terms on one side of the inequality and all the constant terms on the other side.

$6x - 170 \geq 9x - 425$

$- 170 \geq 3x - 425$

$255 \geq 3x$

Step 4: Use inverse operations and the properties of inequality to finish solving.

$255 \geq 3x$

$85{^\circ} \geq x$

Step 5: Graph your answer on a number line.

Step 6: Check your shading.

Choose any value in the shaded area.

Substitute $x = 84$

$2(84) + 4\left( \left( 84 \right) - 42.5 \right) \geq 9(84) - 425$

$2\left( 84 \right) + 4\left( 41.5 \right) \geq 9\left( 84 \right) - 425$

$168 + 166 \geq 756 - 425$

$334 \geq 331$

Step 1: Use the distributive property, if needed.

$1.5\left( 0.5x - 0.5 \right) \leq 0.5x + 2(0.5x - 1.5)$

$0.75x - 0.75 \leq 0.5x + x - 3$

Step 2: Collect the like terms on each side of the inequality, if needed.

$0.75x - 0.75 \leq \color{#A80000}{0.5x + x} - 3$

$0.75x - 0.75 \leq \color{#A80000}{1.5x} - 3$

Step 3: Use the properties of inequality to collect all the variable terms on one side of the inequality and all the constant terms on the other side.

$0.75x - 0.75 \leq 1.5x - 3$

$- 0.75 \leq 0.75x - 3$

$2.25 \leq 0.75x$

Step 4: Use inverse operations and the properties of inequality to finish solving.

$2.25 \leq 0.75x$

$3 \leq x$

Step 5: Graph your answer on a number line.

Step 6: Check your shading.

Choose any value in the shaded area.

Substitute $x = 4$

$1.5\left( 0.5(4) - 0.5 \right) \leq 0.5(4) + 2(0.5(4) - 1.5)$

$1.5\left( 2 - 0.5 \right) \leq 0.5\left( 4 \right) + 2\left( 2 - 1.5 \right)$

$1.5\left( 1.5 \right) \leq 0.5\left( 4 \right) + 2\left( 0.5 \right)$

$2.25 \leq 2 + 1$

$2.25 \leq 3$

Step 1: Use the distributive property, if needed.

$- 5w - 1 \lt - 2(3w - 4)$

$- 5w - 1 \lt - 6w + 8$

Step 2: Collect the like terms on each side of the inequality, if needed.

There are no like terms to combine on either side.

Step 3: Use the properties of inequality to collect all the variable terms on one side of the inequality and all the constant terms on the other side.

$- 5w - 1 \lt - 6w + 8$

$w - 1 \lt 8$

$w \lt 9$

Step 4: Use inverse operations and the properties of inequality to finish solving.

The inequality was solved in the previous step.

Step 5: Graph your answer on a number line.

Step 6: Check your shading.

Choose any value in the shaded area.

Substitute $x = 8$

$- 5(8) - 1 \lt - 2(3(8) - 4)$

$- 5\left( 8 \right) - 1 \lt - 2\left( 24 - 4 \right)$

$- 5\left( 8 \right) - 1 \lt - 2\left( 20 \right)$

$- 40 - 1 \lt - 40$

$- 41 \lt - 40$