Graph solutions on a number line

1. The circle is open (hollow). That means the boundary value 3 is not included, so the symbol must be < or > — no "or equal to".
2. The shading goes to the right, toward bigger numbers. So the solutions are numbers greater than 3.
3. Put the two facts together: x > 3.

Answer: x > 3

Check: 5 is in the shaded part, and 5 > 3 is true ✓. 1 is not shaded, and 1 > 3 is false ✓. 3 sits inside the open circle, and 3 > 3 is false ✓.

Tip: Decide the circle first (open → < or >, closed → \le or \ge), then the direction (right → greater, left → less).

1. Circle: the symbol \le means "less than or equal to", so -1 itself is a solution. An included boundary gets a closed (filled) circle.
2. Direction: numbers less than -1 (like -2, -3, -10) are to the left of -1 on the number line.
3. So: closed circle at -1, shaded to the left.

Answer: A closed circle at -1, shaded to the left.

Check: -1 \le -1 is true, so the circle must be filled ✓. -3 \le -1 is true and -3 is to the left ✓. 0 \le -1 is false and 0 is to the right (unshaded) ✓.

Tip: The little line under \le and \ge is your reminder to "fill in" the circle.

1. Boundary: the circle sits at -4.
2. Circle type: the circle is closed (filled in), so -4 itself is part of the solution. That means the symbol includes "or equal to": \le or \ge.
3. Direction: the shading and arrow go to the left, toward smaller numbers, so the values are less than -4.
4. Put it together: less than or equal to -4.

Answer: x \le -4

Check: Pick a shaded number, -6: is -6 \le -4? Yes ✓. Pick an unshaded number, 0: is 0 \le -4? No ✓. The boundary -4 \le -4 is true, matching the closed circle.

Tip: Read a number-line graph in this order: number → circle (closed = \le/\ge, open = </>) → direction (left = less, right = greater).

1. Boundary: the number in the inequality is -2.5, which sits halfway between -3 and -2.
2. Circle type: the symbol > is strict — -2.5 is not included — so the circle is open.
3. Direction: "greater than" means the solutions are to the right of -2.5 (toward -2, 0, 5, …).

Answer: an open circle at -2.5 with shading to the right.

Check: Pick a number on the shaded side, 0: 0 > -2.5 ✓. Pick a number on the other side, -3: -3 > -2.5 is false ✓ (-3 is further left, so it is smaller). The boundary -2.5 > -2.5 is false, which is why the circle is open.

Tip: For negative numbers, "greater" still means "to the right". -2 is greater than -2.5 because it is to the right on the number line.

1. "At least 48" means 48 or more. A rider who is exactly 48 inches tall is allowed, and anyone taller is too: h \ge 48
2. Circle: \ge includes 48, so the circle at 48 is closed.
3. Direction: "or more" means the shading goes to the right.
4. Graph A has a closed circle at 48 with shading to the right.

Answer: Graph A

Check: A 50-inch rider (50 \ge 48 ✓) is in the shaded part of A. A 46-inch child (46 \ge 48 is false) is not shaded ✓. A child of exactly 48 inches can ride, and the closed circle shows that ✓.

Tip: Learn the phrase pairs: at least / no less than / minimum → \ge; at most / no more than / maximum → \le; more than / over → >; less than / under / below → <.

1. Add 3 to both sides: 2x < 8
2. Divide both sides by 2 (positive, no flip): x < 4
3. Circle: < is strict, so the circle at 4 is open.
4. Direction: "less than" → shade left.

Answer: graph C — open circle at 4, shaded left.

Check: Test x = 0 (shaded): 2(0) - 3 = -3 < 5 ✓. Test x = 6 (unshaded): 12 - 3 = 9 < 5 is false ✓. Boundary: 2(4) - 3 = 5, and 5 < 5 is false, so 4 is correctly excluded.

Tip: Eliminate in two passes — first by circle type (cross off closed circles for <), then by direction with a test point.

1. The shading is between two points, so this is a compound ("between") inequality: (left number) ? x ? (right number).
2. Left end, -2: closed circle → -2 is included → use \le: -2 \le x.
3. Right end, 3: open circle → 3 is not included → use <: x < 3.
4. Combine: -2 \le x < 3.

Answer: -2 \le x < 3

Check: x = -2 is filled in, and -2 \le -2 < 3 is true ✓. x = 3 is hollow, and 3 < 3 is false ✓. x = 0 is shaded, and -2 \le 0 < 3 ✓.

Tip: Read a segment from left to right and match each circle to its own symbol: closed → \le, open → <.

1. The shading lies between -3 and 4, so the inequality has the form -3 \;?\; x \;?\; 4.
2. Left end: the circle at -3 is open, so -3 is not included: -3 < x.
3. Right end: the circle at 4 is closed, so 4 is included: x \le 4.
4. Combine: -3 < x \le 4.

Answer: -3 < x \le 4

Check: 0 is shaded: -3 < 0 \le 4 ✓. 4 is included (closed): -3 < 4 \le 4 ✓. -3 is excluded (open): -3 < -3 is false ✓. 5 is unshaded: 5 \le 4 is false ✓.

Tip: In a "between" inequality, the symbol next to each number comes from the circle at that number: open → <, closed → \le.

1. Read the graph: closed circle at 30, shading to the left → x \le 30. The value 30 is included, and anything smaller is allowed.
2. Translate each situation:
   • Speed no faster than 30 mph → s \le 30 ✓ (30 is allowed).
   • At least 30 → \ge 30 (shades right).
   • Less than 30 → < 30 (open circle).
   • More than 30 → > 30 (open circle, shades right).
3. Only the speed-limit situation matches x \le 30.

Answer: A driver may go no faster than the 30 mph speed limit.

Check: Driving exactly 30 mph is legal (closed circle ✓); 25 mph is legal (shaded ✓); 35 mph is not (unshaded ✓).

Tip: "No faster than", "no more than", "at most" and "maximum" all include the boundary and shade left (\le).

1. Solve: divide both sides by -3 and flip the sign: x < \frac{12}{-3} \quad\Rightarrow\quad x < -4
2. Correct graph: open circle at -4, shading to the left.
3. Compare with Dana's graph:
   • Boundary -4: ✓ correct.
   • Open circle: ✓ correct (< is strict).
   • Shading to the right: ✗ — that shows x > -4, which is what you get if you do not flip.

Answer: She did not reverse the inequality sign, so she shaded the wrong direction; the shading should go to the left.

Check: Test x = 0 (shaded on Dana's graph): -3(0) = 0 > 12 is false, so 0 should not be shaded. Test x = -5: 15 > 12 ✓, so the left side is the real solution.

Tip: A correct boundary does not mean a correct graph. Always test one shaded number in the original inequality.

1. The graph: open circle at 3, shaded left → x < 3.
2. Solve each choice:
   • 2x + 1 < 7: subtract 1 → 2x < 6; divide by 2 → x < 3 ✓
   • 3x \le 9: x \le 3 — closed circle, not open ✗
   • -x < -3: divide by -1, flip → x > 3 ✗
   • x + 2 < 1: subtract 2 → x < -1 ✗
3. Only 2x + 1 < 7 has the solution x < 3.

Answer: 2x + 1 < 7

Check: x = 0 is shaded: 2(0) + 1 = 1 < 7 ✓. x = 3 is excluded: 7 < 7 is false ✓. x = 4 is unshaded: 9 < 7 is false ✓.

Tip: In graph-to-inequality items, the symbol type eliminates choices fast: an open circle rules out every \le or \ge choice before you solve anything.

1. Distribute: 3x - 3 \ge 5x + 7.
2. Subtract 5x from both sides: -2x - 3 \ge 7.
3. Add 3: -2x \ge 10.
4. Divide by -2 and flip: x \le -5.
5. Circle: \le → closed at -5. Direction: shade left.

Answer: graph D — closed circle at -5, shaded left.

Check: Boundary x = -5: left 3(-6) = -18, right 5(-5) + 7 = -18, equal ✓. Test x = -6: 3(-7) = -21 \ge -23 ✓ (shaded). Test x = 0: -3 \ge 7 false ✓ (unshaded).

Tip: Substituting the boundary into the original inequality and getting two equal sides is a fast way to confirm where the circle belongs.

1. Subtract 2 from both sides: -3x < -9
2. Divide by -3 and flip: x > 3
3. The solution set is every number greater than 3 — an open circle at 3, shaded right. Now check each choice:
   • -4: not greater than 3.
   • 2.5: not greater than 3.
   • 3: equal, but the inequality is strict, so not included.
   • 4.5: greater than 3 ✓.

Answer: 4.5

Check: 2 - 3(4.5) = 2 - 13.5 = -11.5, and -11.5 < -7 ✓. And x = 3 gives 2 - 9 = -7, and -7 < -7 is false.

Tip: Picture the graph: open circle at 3, arrow to the right. Only numbers to the right of 3 count.

1. The graph: closed circle at -1, shaded right → x \ge -1.
2. Solve each choice:
   • 4 - 3x \le 7: subtract 4 → -3x \le 3; divide by -3, flip → x \ge -1 ✓
   • 4 - 3x \ge 7: -3x \ge 3 → x \le -1 ✗ (shades left)
   • 3x + 4 \ge 7: 3x \ge 3 → x \ge 1 ✗ (boundary 1)
   • 4 - 3x < 7: -3x < 3 → x > -1 ✗ (open circle)
3. Only 4 - 3x \le 7 matches.

Answer: 4 - 3x \le 7

Check: x = 0 (shaded): 4 \le 7 ✓. x = -1 (closed circle): 4 + 3 = 7 \le 7 ✓. x = -2 (unshaded): 10 \le 7 is false ✓.

Tip: When the graph shades right but the inequality says \le, expect a negative coefficient — the flip turns \le into \ge.

1. New temperature = current temperature + rise = -3 + t.
2. "At or below 5" means \le 5: -3 + t \le 5
3. Add 3 to both sides: t \le 8
4. Circle: \le includes 8 → closed. Direction: less than → shade left.
5. Graph A shows a closed circle at 8 shaded left.

Answer: Graph A

Check: A rise of 8 degrees gives -3 + 8 = 5^\circ\text{F}, which is allowed (closed circle ✓). A rise of 9 degrees gives 6^\circ\text{F}, too warm (unshaded ✓). A rise of 2 degrees gives -1^\circ\text{F}, fine (shaded ✓).

Tip: In context problems, write the inequality in words first ("now + rise ≤ limit"), then fill in the numbers. That keeps the operation right.

1. Distribute \frac{2}{3} to both terms: \frac{2}{3} \cdot 6 = 4. \frac{2}{3}x + 4 > x + 3
2. Subtract \frac{2}{3}x from both sides (keeps the x-coefficient positive): 4 > \frac{1}{3}x + 3
3. Subtract 3: 1 > \frac{1}{3}x
4. Multiply by 3 (positive, no flip): 3 > x, \text{ which means } x < 3
5. Open circle at 3 (strict), shaded left. That is Graph C.

Answer: Graph C

Check: x = 0: left \frac{2}{3}(6) = 4, right 3; 4 > 3 ✓ (0 is shaded in C). x = 6: left 8, right 9; 8 > 9 is false ✓. Boundary x = 3: left 6, right 6; 6 > 6 is false, so the circle is open ✓.

Tip: If you subtract x instead and get -\frac{1}{3}x > -1, you must flip when you multiply by -3 — you still get x < 3. Either path works if you follow the rule.

1. Read the graph: closed circle at 5, shaded left → x \le 5. Boundary: 5.
2. At the boundary, the two sides are equal: 10 - a(5) = -5
3. Subtract 10: -5a = -15.
4. Divide by -5: a = 3.
5. Direction check: with a = 3, 10 - 3x \ge -5 → -3x \ge -15 → divide by -3, flip → x \le 5 ✓ (shaded left, closed).

Answer: a = 3

Check: x = 5: 10 - 15 = -5 \ge -5 ✓. x = 0 (shaded): 10 \ge -5 ✓. x = 6 (unshaded): 10 - 18 = -8 \ge -5 is false ✓.

Tip: After finding a constant from the boundary, always solve forward once. A wrong sign on the constant gives the right boundary but the wrong direction.

1. Distribute -2: -2x - 2 < 4 - x.
2. Add x to both sides: -x - 2 < 4.
3. Add 2 to both sides: -x < 6.
4. Divide by -1 and flip: x > -6.
5. Correct graph: open circle at -6, shaded right — that is Lena's graph. Omar's graph (shaded left) shows x < -6, the result of not flipping.

Answer: Lena's graph is correct; Omar forgot to reverse the inequality sign.

Check: Test x = 0 (shaded only in Lena's graph): -2(1) = -2 < 4 ✓. Test x = -8 (shaded only in Omar's graph): -2(-7) = 14 and 4 + 8 = 12; 14 < 12 is false ✓.

Tip: When two graphs share the same boundary and circle, a single test point on one side settles which direction is right.